CONNECTION BLOCKING IN LATTICE QUOTIENTS OF CONNECTED LIE GROUPS By Mohammadreza Bidar A DISSERTATION to Michigan State University in partial fulfillment of the requirements Submitted for the degree of Mathematics – Doctor of Philosophy 2018 CONNECTION BLOCKING IN LATTICE QUOTIENTS OF CONNECTED LIE GROUPS ABSTRACT By Mohammadreza Bidar Finite blocking is an interesting concept originating as a problem in billiard dynamics and later in the context of Riemannian manifolds. Let (M, g) be a complete connected, infinitely dierentiable Riemannian manifold. To block a pair of points m1, m2 2 M is to find a finite set B ⇢ M \{m1, m2} such that every geodesic segment joining m1 and m2 intersects B. B is called a blocking set for the pair m1, m2 2 M. The manifold M is secure if every pair of points in M can be blocked. M is uniformly secure if the cardinality of blocking sets for all pairs of points in M has a (finite) upper bound. The main blocking conjecture states that a closed Riemannian manifold is secure if and only if it is flat. Gutkin [15] initiated a similar study of blocking properties of quotients G/ of a connected Lie group G by a lattice ⇢ G. Here the connection curves are the orbits of one parameter subgroups of G. To block a pair of points m1, m2 2 M is to find a finite set B ⇢ M \ {m1, m2} such that every connection curve joining m1 and m2 intersects B. The lattice quotient M = G/ is connection blockable if every pair of points in M can be blocked, otherwise we call it non-blockable. The corresponding main blocking conjecture states that M = G/ is blockable if and only if its universal cover ˜G = Rn, i.e. M is a torus. In this dissertation we investigate blocking properties for two classes of lattice quotients, which are lattice quotients of semisimple and solvable Lie groups. According to the Levi decomposition, every connected Lie group G is a semidirect product of a solvable Lie group R, and a semisimple Lie group S. A Lie group G = RoS satisfies Raghunathan’s condition if the kernel of the action of S on R has no compact factors in its identity component. For a such Lie group G, if quotients of R are non-blockable then quotients of G are also non-blockable. The special linear group SL(n, R) is a simple Lie group for n > 1. Let Mn = SL(n, R)/, where = SL(n, Z) is the integer lattice. We focus on M2 and show that the set of blockable pairs is a dense subset of M2 ⇥ M2, and we use this to conclude manifolds Mn are non-blockable. Next, we review a quaternionic structure of SL(2, R) and a way for making cocompact lattices in this context. We show that the obtained lattice quotients are not finitely blockable. In the context of solvable Lie groups, we study lattice quotients of Sol. Sol is a unimodular solvable Lie group, with the left invariant metric ds2 = e2zdx2 + e2zdy2 + dz2, and is one of the eight homogeneous Thurston 3-geometries. We prove that all quotients of Sol are non-blockable. In particular, we show that for any lattice ⇢ Sol, the set of non-blockable pairs is a dense subset of Sol/ ⇥ Sol/. Copyright by MOHAMMADREZA BIDAR 2018 ACKNOWLEDGEMENTS First and foremost I want to thank my advisor Benjamin Schmidt. It has been an honor for me to be his PhD student. During my PhD career he introduced me to several interesting research problems, and the potential solution plans which turned out to be very helpful. I appreciate all his contributions of time and ideas to make my PhD experience productive and stimulating. The joy and enthusiasm he has for his research has alway been motivational for me, even during tough times I have had in the PhD pursuit. I would like to thank Thomas Parker for his valuable advices during my PhD career, and his recommendations for improving my dissertation writing. I also would like to thank Rajesh Kulkarni and Xiaodong Wang for reviewing my dissertation and being in my dissertation defense committee. Rajesh has always been a great help for answering algebra questions I have had during my thesis research. During my time at MSU I have had the pleasure of working with many nice and friendly people in Math Department. I would like to thank my teaching colleagues and calculus supervisors for their awesome supervision, friendly communication and professional care: Gabriel Nagy, Tsvetanka Sendova, Rachael Lund, Ryan Maccombs and Andrew Krause. I would also like to thank Math Department of MSU and Michigan State University for being fully supportive, and for making my time at MSU wonderful and productive. Lastly, I would like to thank my family for all their love and encouragement. For my parents who raised me with a love of science and supported me in all my pursuits, even though they have not been here beside me during my PhD career. v TABLE OF CONTENTS . . . . . KEY TO SYMBOLS . . INTRODUCTION . CHAPTER 1 CHAPTER 2 PRELIMINARIES . 2.1 Lie Theory Background . . 2.2 Connection Blocking in Lattice Quotients of Connected Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 . . . . . . . . 18 CHAPTER 3 CONNECTION BLOCKING IN QUOTIENTS OF SOL . . . . . . . . . . . 24 . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1 3.2 Lattices in Sol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Blocking Property of Sol Quotient Spaces . . . . . . . . . . . . . . . . . . . . . . 29 CHAPTER 4 CONNECTION BLOCKING IN SEMISIMPLE LATTICE QUOTIENTS . . 34 4.1 One Parameter Families of SL(2, R) and Modified Times . . . . . . . . . . . . . . 34 4.2 Blocking Properties of Mn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 . . . . . . . . . . . . . . . 42 4.3 Blocking Property and Cocompact Lattices of SL(2, R) Sol and One Parameter Subgroups . . . . . . CHAPTER 5 CONCLUDING REMARKS AND PROPOSED PROBLEMS FOR FUR- APPENDICES . . . THER RESEARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.1 Connection Blocking Problems in Other Lattice Quotients . . . . . . . . . . . . . . 49 5.2 Behavior of Exponential Map Near Singularities . . . . . . . . . . . . . . . . . . . 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 APPENDIX A Semidirect Product and Semidirect Sum . . . . . . . . . . . . . . . . . 55 APPENDIX B Levi Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 APPENDIX C Haar Measure APPENDIX D Arithmetic Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 . . . . . . . . . . . . . BIBLIOGRAPHY . . . vi KEY TO SYMBOLS o — Sign of the semidirect product of groups (the normal sub group is to the left) — Sign of the semidirect sum of algebras (the ideal is to the left) — Adjoint representation of a Lie Group — Adjoint representation of a Lie Algebra — Radical of a set X — The identity component of a lie group G — Set of smooth functions defined on G — Radical of a lie group G + Ad ad Sqrt(X) G C1(G) Rad G CommG(⇤) — Commensurator of a discrete subgroup ⇤ in G F-rank(G) — Rank of a Lie group G over a field F der g rad g Im,n GL(n, F) SL(n, F) O(n) SO(n) SO(m, n) Ha,b F Nred SL(Ha,b F ) nT(x, y) mT(x, y) — Lie algebra of derivations of g — Radical of a Lie algebra g — Diagonal n ⇥ n matrix where its diagonal consists of m 1’s followed by n (-1)’s — Group of invertible n ⇥ n matrices over a field F — Linear special group (subgroup of GL(n, F) whose elements have determinant 1) — Orthogonal group of n ⇥ n matrices (over R) — Special orthogonal group of n ⇥ n matrices (over R) — Special orthogonal group of n ⇥ n matrices g with gT Im,ng = Im,n — Algebra of quaternions over a field F — Reduced norm — Subalgebra of quaternions with norm one, over a field F — Number of geodesic segments joining x, y of length  T — Number of geodesic segments connecting x, y of length  T vii CHAPTER 1 INTRODUCTION The theme of finite blocking originates from a problem in the Leningrad Mathematical Olympiad worded as follows. The president and a terrorist are moving in a rectangular room. The terrorist intends to shoot the president with his ‘magic gun’ whose bullets bounce o the walls perfectly elastically: the angles of incidence and reflection are equal. Presidential protection detail consists of superhuman body guards. They are not allowed to be where the president or the terrorist are located, but they can be anywhere else, changing their locations instantaneously, as the president and the terrorist are moving about the room. Their task is to put themselves in the way of terrorist’s bullets shielding the president. The problem asks how many body guards suce. To translate this problem into a mathematical setting, let ⌦ be a bounded plane domain. For arbitrary points p, q 2 ⌦ let (p, q) be the family of billiard orbits in ⌦ connecting these points. Body guards correspond to b1, ..., bN 2 ⌦ \ {p, q} such that every 2 (p, q) passes through one of these points. If for any p, q 2 ⌦ there is a blocking set B = B(p, q) = b1, ..., bN then the domain is uniformly secure. The minimal possible N is then the blocking number of ⌦. The Olympiad problem is to show that a rectangle is uniformly secure and to find its blocking number. The solution leads to a problem in plane geometry based on two facts: 1. A rectangle tiles the Euclidean plane under reflections; 2. The torus T2 = R2/Z2 is uniformly secure, where the role of billiard orbits is played by the images of straight lines under the projection R2 ! T2. A blocking set in the torus is the set of midpoints of all joining segments: It comprises at most 4 points. A blocking set in the rectangle is also the set of midpoints of all joining billiard orbits. It comprises at most 16 = 4◊4 points where the factor 4 is due to the 4 copies of the rectangle needed to tile the torus. In the context of planar geometry, we may be ask a similar question for polygon billiards. For 1 a solution we need to find out which plane polygons are secure. This problem first appeared in the literature in Hiemer and Snurnikov [22]. A polygon is rational if its corners have ⇡-rational angles. It is claimed in [22] that all rational polygons are secure. Gutkin studied the security of translation surfaces [16, 17] and proved that the regular n-gon is secure if and only if n = 3, 4, 6 [12]. Since all regular polygons are rational, this disproves the claim in [22]. The work of Gutkin [13] contains related results on the security of rational polygons, but a solution to the general case remains elusive [14]. The billiard orbits in the rectangle and the straight lines in the torus are examples of geodesics in Riemannian manifolds. The original Olympiad problem expanded into the subject of Riemannian security. To study the security of Riemannian manifolds, first we need to define the problem in mathemat- ical terms. Let (M, g) be a complete connected, infinitely dierentiable Riemannian manifold. For a pair of (not necessarily distinct) points m1, m2 2 M let (m1, m2) be the set of geodesic segments joining these points. A set B ⇢ M \ {m1, m2} is blocking if every 2 (m1, m2) intersects B. The pair m1, m2 is secure if there is a finite blocking set B = B(m1, m2). A manifold is secure if all pairs of points are secure. If there is a uniform bound on the cardinalities of blocking sets, the manifold is uniformly secure and the best possible bound is the blocking number. Now, the first question naturally arising is which Riemannian manifolds are secure. If we focus on closed Riemannian manifolds, there is the following conjecture as stated by Burns-Gutkin and Lafont-Schmidt [5, 25]: Conjecture 1.1. A closed Riemannian manifold is secure if and only if it is flat. Conjecture 1.1 says that flat manifolds are the only secure manifolds. This has been verified for several special cases: • Flat manifolds are uniformly secure, and the blocking number depends only on their dimension (Gutkin-Schroeder [18, 12]). In fact, they are also midpoint secure, i.e., the midpoints of 2 connecting geodesics yield a finite blocking set for any pair of points (Gutkin-Schroeder and Bangert-Gutkin)[18, 3, 12]. • A manifold without conjugate points is uniformly secure if and only if it is flat (Burns-Gutkin and Lafont-Schmidt[5, 25]). • A compact locally symmetric space is secure if and only if it is flat (Gutkin-Schroeder [18]). • The generic manifold is insecure (Gerber-Ku and Hebda-Ku[8, 20]). • Conjecture 1.1 holds for compact Riemannian surfaces with genus bigger or equal than one (Bangert-Gutkin [3]). • Any Riemannian metric has an arbitrarily close, insecure metric in the same conformal class (Hebda-Ku [20]). To have a better insight of the security concept, we present the proof of Proposition 2 in [18] to show that the flat torus is uniformly (midpoint) secure: Proposition 1.2. The flat torus Tn = Rn/Zn is uniformly secure and the blocking number is 2n. Proof. Let o 2 Tn be the origin. Let x 2 Tn be an arbitrary point. By homogeneity, it suces to prove that the pair (o, x) is blockable with 2n points. Let G(o, x) be the set of all geodesics connecting the origin o, and x. There is a one-to-one correspondence between the geodesics 2 G(o, x) and the straight segments ˜x+z 2 Rn connecting the origin O 2 Rn with the points x + z, z 2 Zn. Let x+z 2 G(o, x) be the corresponding connecting geodesic. If p : Rn ! Tn is the projection, then x+z = p( ˜x+z). The midpoint of the segment ˜x+z is x 2 2 Rn. Set ˜F(x) = { x 2 : z 2 Zn}. Then the set F(x) = p( ˜F(x)) ⇢ Tn is finite, and |F(x) = 2n|. Thus, 2n points suce to block any 2 G(o, x). On the other hand, for a typical x, we cannot block G(o, x) with less than 2n points. We leave the verification of this to the reader. ⇤ 2 + z 2 + z By the Bieberbach theorem, every closed flat Riemannian manifold M is finitely covered by It is a flat torus (For statement and proof of the Bieberbach theorem see Wolf [37, p.100]). 3 straightforward to see that if the finite cover of a Riemannian manifold is uniformly secure, then the manifold is uniformly secure. Therefore, Proposition 1.2 implies the following corollary: Corollary 1.3. Every flat closed Riemannian manifold is uniformly secure. Gutkin [15] initiated the study of blocking properties of lattice quotients of connected Lie groups. In this context, he speaks of finite blocking instead of security. Let G be a connected Lie group, and let ⇢ G be a lattice. Connection curves of the lattice quotient M = G/ are the orbits of one parameter subgroups of G. To block a pair of points m1, m2 2 M is to find a finite set B ⇢ M \ {m1, m2} such that every connection curve joining m1 and m2 intersects B. If every pair of points in M can be blocked, M is called connection blockable, or simply blockable, otherwise it is called non-blockable. A counterpart of Conjecture 1 for lattice quotients is as follows: Conjecture 1.4. Let G be a connected Lie group with the universal cover ˜G, ⇢ G a lattice, and let M = G/. Then M is blockable if and only if ˜G = Rn, i.e. M is a torus. To start working on this conjecture, it would be helpful to consider solvable and semisimple Lie groups first. Solvable Lie groups (resp. Lie algebras) and the semisimple Lie groups (resp. Lie algebras) form two large and generally complementary classes. Every connected Lie group G is a semidirect product of a solvable Lie group R, and a semisimple Lie group S (Theorem B.4) which is called the Levi decomposition. A connected Lie group G satisfies Raghunathan’s condition if the kernel of the action of S on R has no compact factors in its identity component. For such Lie groups G, if lattice quotients of R are non-blockable, then lattice quotients of G are also non-blockable. In addition, we will show in Proposition 2.18 that if nilradical of G is not abelian, then lattice quotients of G are non-blockable. Gutkin in [15] establishes Conjecture 1.4 for lattice quotients of nilpotent Lie groups. Such spaces are called nilmanifolds. He starts with the connection blocking problem in Heisenberg manifolds. He then studies connection blocking in two-step nilmanifolds, and then extends the results to an arbitrary nilmanifold. 4 The Lie group H is the Lie subgroup of GL(3, R) defined by 1 x z 0 1 y 0 0 1 H =8>>>>><>>>>>: ©≠≠≠≠´ : x, y, z 2 R9>>>>>=>>>>>; ™ÆÆÆƨ . A Heisenberg manifold is a quotient of H by a cocompact lattice. Gutkin’s main result can be summarized in the following propositions (See Theorems 1 and 2 in [15]). Proposition 1.5. Let M be a three-dimensional Heisenberg manifold. Then i) A pair of points in M is blockable if and only if it is midpoint blockable. ii) For every x 2 M, the pair (x, x) is blockable. iii) The set of blockable pairs of points is a dense countable union of closed submanifolds of positive codimension in M ⇥ M. iv) In particular iii) implies that almost all pairs of points (x, y)2 M ⇥ M are non-blockable. Proposition 1.6. Let M be a nilmanifold of dimension n . Then the following statements are equivalent: i) M is connection blockable; ii) M is midpoint blockable; iii) ⇡1(M) = Zn; iv) M is a topological torus; v) M is uniformly blockable and the blocking number depends only on its dimension. Every nilpotent Lie groups is solvable; however, solvable Lie groups constitute a much larger class. One the simplest non-nilpotent solvable Lie groups is Sol. Sol is the Lie group of all vectors 5 (x, y, z)2 R3 with the group multiplication(x1, y1, z1)(x2, y2, z2) = (x1+ez1 x2, y1+ez1 y2, z1+z2). Details of the definition and properties of Sol are presented in Chapter 3. We prove Conjecture 1.4 for lattice quotients of the Lie group Sol. We start with a specific class of lattices in Sol: those that are isomorphic to the semidirect product Z2 oA Z, where A 2 SL(2, Z) is a diagonalizable matrix, and r 2 Z acts on Z2 as Ar, so as the multiplication is given by (p1, q1, r1)(p2, q2, r2) = ((p1, q1) + Ar1(p2, q2), r1 + r2) . If P is the eigenvector matrix of A, by Proposition 3.4 the mapping (p, q, r) 7! (P1(p, q), sr) embeds Z2 oA Z into Sol and the image is a cocompact lattice. We then solve the blocking problem for some of these lattices. In Section 3.3 we prove: Theorem 1.7. Let A 2 SL(2, Z) be a matrix with eigenvalues ±{, 1}, where = es for some s , 0. Then there exists P 2 GL(2, R) such that P11 = P22 = 1, and PAP1 =©≠≠´ . 0 0 1™Æƨ Let = (A) = {(P(p, q), sr)|p, q, r 2 Z} be the corresponding lattice in Sol. Let m1 = g1, m2 = g2 be a pair of points in Sol/, and assume g1 If x0 = 0, y0 , 0, or y0 = 0, x0 , 0, then m2 is not blockable from m1. 1 g2 = (x0, y0, z0). Remark. The above theorem basically shows that if two points are on the planes x = c, or y = c, (not having the same y, or x coordinates, respectively) then their corresponding cosets in the quotient space are not blockable. Through Proposition 3.6 and Lemma 3.6 in Chapter 3 we show that all lattices of Sol are isomorphic to the semidirect product lattices presented in Theorem 1.7, and we then prove non- blockability of all quotients of Sol, as stated in the following theorem. Theorem 1.8. All lattice quotients of Sol are non-blockable. In fact, for every lattice in Sol, the set of non-blockable pairs is a dense subset of Sol/ ⇥ Sol/. 6 In the context of semisimple Lie groups, we investigate blocking problem for quotient lattices of SL(n, R). Gutkin in [15] proves the lattice quotient SL(n, R)/SL(n, Z) is not midpoint blockable. For simplicity, we use the following notation throughout the thesis. Notation. For n > 1, Mn denotes the homogeneous space SL(n, R)/SL(n, Z), and when it is clear from the context, denotes SL(n, Z). We prove that Mn and all quotients of SL(n, R) whose lattice is commensurable to SL(n, Z), are non-blockable. Specifically we prove the following theorem. Theorem 1.9. Two elements m1 = g1 and m2 = g2 2 M2 are not finitely blockable from each other if g1 1 g2 2 SL(2, Q). In fact, the set of non-blockable pairs is a dense subset of M2 ⇥ M2. This easily implies the following: Theorem 1.10. Any lattice quotient Mn, n > 2, has infinitely many pairs of non-blockable points. Remark. Lattices 1 and 2 of a Lie group G are commensurable if there exists g 2 G such that the group 1 \ g2g1 has finite index in both 1 and g2g1. By Corollary 2.13, quotient spaces of a Lie group mod two commensurable lattices carry the same blocking property. The Margulis Arithmeticity Theorem (See Morris and Margulis [31, p.92], [29, p.298]), implies every lattice of SL(n, R), n 3 is arithmetic. As a result, a large class of lattices in SL(n, R), n 3 are commensurable to SL(n, Z). In particular, if is a lattice and the subgroup \ SL(n, Z) is of finite index in , then and SL(n, Z) are commensurable. Hence, all lattice quotients SL(n, R)/, for such lattices are non-blockable. Moreover, for every lattice ⇢ SL(n, Q), SL(n, R)/ is non-blockable [29, p.319]. A lattice ⇢ G is called cocompact if the quotient space G/ is compact. SL(2, Z) is the most basic example of a non-cocompact lattice in SL(2, R). Up to commensurability and conjugates, this is the only one that is not cocompact, Morris [31, p.115]. Since lattice quotients of the same Lie group mod conjugate or commensurable lattices have identical blocking property (See Corollary 2.13), we conclude following corollary. 7 Corollary 1.11. For every non-cocompact lattice ⇢ SL(2, R), the quotient space SL(2, R)/ is non-blockable. For SL(2, R), we additionally study the blocking problem for a class of compact lattice quotients. We show that for a large class of cocompact lattices ⇢ SL(2, R), defined using a quaternionic structure, SL(2, R)/ is non-blockable. For any field F, and any nonzero a, b 2 F, Ha,b F is a quaternion algebra, in which the multiplication and the norm function depend on a, b, and SL(1, Ha,b F ) is the subgroup of elements with norm one. See Section 4.3 for the formal mathematical definitions. We specifically prove the following theorem. Theorem 1.12. Let a, b be positive integers such that = SL(1, Ha,b G = SL(1, Ha,b Therefore the lattice quotient G/ is not finitely blockable. R ). If g = x + yi 2 SL(1, Ha,b Z ) is a cocompact lattice of Q ), then g ⇢ G/ is not finitely blockable from m0 = . The organization of the dissertation is as follows. In Chapter 2, we briefly review the require- ments needed to study the blocking problem. Section 2.1 includes Lie theory basic concepts and theorems, as presented in the standard textbooks. In Section 2.2 we review general blocking prop- erties of lattice quotients of a connected Lie group. In Chapter 3 we study the blocking problem in lattice quotients of Sol. In Section 3.1 we derive an explicit formula for Sol’s one parameter subgroups. Section 3.2 introduces the semidirect product lattices in Sol. We describe a group presentation for all lattices in Sol according to Molnár [30] and then prove that all lattices in Sol are conjugate to semidirect product lattices. In Section 3.3, we first prove a few technical lemmas, then proceed to prove Theorems 1.7 and 1.8. In Chapter 4 we study the blocking problem in lattice quotients of SL(n, R), mainly focusing on SL(2, R). In Section 4.1 we describe one parameter subgroups in the Lie group SL(2, R). In Section 4.2, we first prove a technical proposition, then state and prove Theorems 1.9 and 1.10. Section 4.3 presents a quaternionic structure of SL(2, R) and a way for making cocompact lattices in this context. The section concludes with the proof of Theorem 1.12. Chapter 5 includes concluding remarks and problems for further research. Section 5.1 discusses the blocking problem for some other solvable and semisimple Lie groups. In Section 8 5.2 we discuss how the blocking problem in manifolds with conjugate points relates to the behavior of the exponential map near singularities. We then state a conjecture about the number of geodesic segments of certain length, and propose a sketch for a potential proof. 9 CHAPTER 2 PRELIMINARIES This chapter is a brief review of the background needed to study the blocking problem. The first section includes Lie Theory basic concepts and theorems, as presented in standard textbooks. In the second section we introduce the concept of connection blocking in lattice quotients of connected Lie groups and review some general blocking properties for these spaces. 2.1 Lie Theory Background In this section we briefly review the basic concepts of Lie Theory. Proofs of the theorems and detailed discussions can be found in many Lie theory textbooks, for example Abbaspour [1], Hilgert [23], and Hall [19]. See Morris [31] for a detailed discussion about lattices and arithmetic subgroups. 2.1.1 Lie Groups and Lie Algebras By a Lie group G we mean a topological group with a dierentiable structure such that the mapping G ⇥ G ! G given by (x, y)! xy1, x, y 2 G, is dierentiable. It follows that left translations Lg : G ! G, Lg(h) = gh, and right translations Rg : G ! G, Rg(h) = hg, are dieomorphisms. We say that a Riemannian metric on G is left invariant if < U, V >h=< d(Lg)hU, d(Lg)hV > for all g, h 2 G, U, V 2 ThG, that is, if Lg is an isometry. Analogously, we can define a right invariant Riemannian metric. A Riemannian metric on G which is both right and left invariant is said to be bi-invariant. We say that a dierentaible vector field X on a Lie group G is left invariant if dLg(X) = X for all g 2 G. From The left translation Lg one can, for any vector Xe 2 TeG, define a left invariant vector field X on G by Xg = (dLg)(Xe) . 10 X is left invariant since dLg(Xh) = dLg dLh(Xe) = dLgh(Xe) = Xgh . Let g represent the set of left invariant vector fields on G. For a, b 2 R and X,Y 2 g, it is easy to see aX + bY 2 g, which means g is a R-vector space. Recall that the Lie bracket of two vector fields X,Y on G is defined as [X,Y] := XY Y X, which is the same as Lie derivative of Y with respect to X. It’s not dicult to see if X,Y 2 g, so is their Lie bracket [X,Y]. It follows that for a Lie group G as an n-dimensional smooth manifold, g is an n-dimensional vector space, and together with Lie bracket operation [·, ·] it’s called the Lie Algebra of G. Since every left invariant vector field X is identified by Xe, its value at identity, g can also be defined as the tangent space at identity, TeG. Lie bracket of Xe,Ye 2 TeG is then defined by [Xe,Ye] := [X,Y]e. A Lie Algebra homomorphism between two Lie algebras is a linear map that preserves Lie algebra structure. Suppose : G ! H is a Lie group homomorphism, then its dierential at e, d : TeG ! TeH gives a linear map from TeG to TeH. Considering the identification of TeG with g and TeH with h, we state the following theorem without proof. Theorem 2.1. If : G ! H is a Lie group homomorphism, then the induced map d : g ! h is a Lie algebra homomorphism. Remark. A Lie algebra in abstract sense is defined as a vector space g together with a bilinear, antisymmetric map g ⇥ g ! g, (X,Y) 7! [X,Y], called the Lie bracket, satisfying Jacobi identity, i.e. for all X,Y, Z 2 g, [X,[Y, Z]] +[Y,[Z, X]] +[Z,[X,Y]] = 0. A Lie algebra g is abelian if the Lie bracket vanishes identically, that is for all X,Y 2 g, [X,Y] = 0; otherwise it’s called non-abelian. As we saw earlier, every Lie group gives rise to a Lie algebra. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to covering (Lie’s third theorem, see Theorem B.4). This correspondence between Lie groups and Lie algebras allows one to study Lie groups in terms of Lie algebras. 11 2.1.2 One Parameter Subgroups Let G be a Lie group, Xe 2 TeG be a tangent vector at the identity element, and X 2 g be the left invariant vector field generated by Xe. As an exercise one can show that any left invariant vector field on G is complete. So for any g 2 G there is a unique integral curve of X defined on the whole real line R, g : R ! G , so that g(0) = g. We are interested in the special map := e, the integral curve of X that starts at e. It is easy to see that the map = e is a Lie group homomorphism from R to G, i.e. (s + t) = (s)(t) holds for all s, t 2 R. Definition 2.2. A one-parameter subgroup of a Lie group G is a Lie group homomorphism : R ! G, that is is smooth and, (s + t) = (s)(t) holds for all s, t 2 R. So the argument above shows that for any X 2 g (or for any Xe 2 TeG), one can construct a one-parameter subgroup of G. Conversely, for any one-parameter subgroup : R ! G, we must have (0) = e, and thus construct a left-invariant vector field X on G via the vector Xe = €(0) = (d)0✓ d dt◆ 2 TeG . It is not hard to see that dierent vectors in TeG give rise to dierent one-parameter subgroups, and dierent one-parameter subgroups give rise to dierent vectors in TeG. As a consequence, we get one-to-one correspondence between 1. One-parameter subgroups of G, 2. Left invariant vector fields on G, 3. Tangent vectors at e 2 G. So we have three dierent descriptions of the Lie algebra g. 12 2.1.3 The Exponential Map For any X 2 g, let X be the one-parameter subgroup G corresponding to X. Definition 2.3. The exponential map of G is the map exp : g ! G, X 7! X(1) . Since ˜(s) = X(ts) is the one parameter subgroup corresponding to tX, we have exp(tX) = X(t) . Note that the zero vector 0 2 TeG generates the zero vector field on G, whose integral curve through constant e is the constant curve. So exp(0) = e. The exponential map exp : g ! G is a local dieomorphism near 0 and it’s dierential at 0 is the identity map. i.e. (d exp)0 = Id . The exponential map is natural, which means for any Lie group homomorphism : G ! H, the diagram g expg G d h exph H (2.1.1) is commutative, i.e. expg = exph d. 2.1.4 Linear Lie Groups Recall that M(n, R), the set of all n ⇥ n real matrices, is dieomorphic to Rn2. Definition 2.4. A linear Lie group, or matrix Lie group, is a submanifold of M(n, R) which is also a Lie group, with group structure the matrix multiplication. Let’s begin with the “largest” linear Lie group, the general linear group GL(n, R) = {X 2 M(n, R) | det X , 0} . 13 Since the determinant map is continuous, GL(n, R) is open in M(n, R) and thus a submanifold. Moreover, GL(n, R) is closed under the group multiplication and inversion operations, so it is a Lie group. Obviously GL(n, R) is an n2-dimensional non-compact Lie group, and it is not connected. In fact, it consists of exactly two connected components which are GL+(n, R) = {X 2 M(n, R) | det X > 0} , and GL(n, R) = {X 2 M(n, R) | det X < 0} . The fact that GL(n, R) is an open subset of M(n, R) Rn2 also implies that the Lie algebra of GL(n, R), as the tangent space at e = In, is the set M(n, R) itself, i.e. gl(n, R) = {A | A is an n ⇥ n real matrix} . Using the coordinate system computations, it turns out that the Lie bracket operation on g is the matrix commutator, that is for all A, B 2 g [A, B] = AB BA . Given any A 2 gl(n, R), we can define the matrix exponential A3 An 3! + · · · + n! + · · · . A2 2! + It is easy to check that the series converges, and eA = In + A + esAet A = e(s+t)A . Notice that e0A = In, and (et A)1 = et A. parameter subgroup of GL(n, R). Since d gl(n, R)! GL(n, R) is dtt=0 In particular, et A 2 GL(n, R). So et A is a one- et A = A, we conclude that the exponential map A2 2! + A3 3! + · · · + An n! + · · · . exp(A) = In + A + 14 The exponential map is not surjective, not even to GL+(n, R). A subgroup H of a Lie group G is called a Lie subgroup if it is a Lie group (with respect to the induced operation), and the inclusion map ◆H : H ,! G is an immersion (and therefore a Lie group homomorphism). Suppose H is a Lie subgroup of G, and let h be the Lie algebra of H. Since ◆ : H ,! G is an immersion and is a Lie group homomorphism, d◆H : h ! g is injective and is a Lie algebra homomorphism. So we think of h as a Lie subalgebra, i.e. a linear subspace that is closed under the Lie bracket of g. Note that a one-parameter subgroup of H is automatically a one-parameter subgroup of G (with initial vector in TeH), so the exponential map expH : h ! H is exactly the restriction of expG : g ! G onto h. The following well known theorem is very useful to determine the Lie algebra of a lie subgroup. Theorem 2.5. Suppose H is a Lie subgroup of G. Then as a Lie subalgebra of g, h = {X 2 g | expG(tX)2 H for all t 2 R} . The special linear group. The special linear group is defined as SL(n, R) = {X 2 GL(n, R) | det X = 1} . It is easy to see that SL(n, R) is a subgroup, and a n2 1 dimensional submanifold of GL(n, R). It follows that SL(n, R) is a (connected non-compact) Lie subgroup of GL(n, R). To determine its Lie algebra sl(n, R), first notice that det eA = eTr(A). So for an n⇥ n matrix A, eA 2 SL(n, R) if and only if Tr(A) = 0. We conclude that sl(n, R) = {A 2 gl(n, R) | Tr(A) = 0} . 2.1.5 Lattices By a discrete subgroup ⇤ of a Lie group G we mean a Lie subgroup of G such that it has discrete topology as a topological subspace. 15 Commensurability. When studying discrete subgroups and lattices, we usually wish to ignore minor dierences that come from passing to a finite-index subgroup. This leads us to the concept of commensurability. Definition 2.6. Two subgroups H1 and H2 of the same group G are said to be (strictly) commen- surable if H1 \ H2 is a finite index subgroup of both H1 and H2. A basic exercise in group theory shows that the intersection of two finite-index subgroups is again a finite index subgroup. It follows that being commensurable is an equivalence relation on the set of subgroups of a given group. If one wishes to go from geometry to algebra, there are some modifications that we need to consider. In the context of blocking problem the following definition is the appropriate modification we need to make. Definition 2.7. Two subgroups H1 and H2 of a group G are said to be weakly commensurable if there is an element of g 2 G such that H1 and gH2g1 are strictly commensurable. It is straightforward to see that being weakly commensurable is again an equivalence relation on the subgroups of a given group G. Two discrete subgroups of a Lie group which are weakly commensurable have very similar geometric structure; this leads us to the following definition: Definition 2.8. An element g 2 G commensurates ⇤ if g⇤g1 is commensurable to ⇤. Let CommG(⇤) := {g 2 G | g commensurates ⇤} . This is called the commensurator of ⇤. CommG(⇤) contains the normalizer NG(⇤) of G, and is sort of generalized normalizer of G. Indeed, CommG(⇤) can be thought of as the stabilizer of the commensurability class of ⇤ for the conjugacy action of G on the set of commensurability classes of its subgroups. For example, if G = SL(n, R) and ⇤ = SL(n, Z), then NG(⇤) is commensurable to ⇤, but CommG(⇤) contains SL(n, Q); so CommG(⇤) is dense in G, even though NG(⇤) is discrete (see Morris [31, p.49]). 16 Remark. In the context of blocking problem, when we refer to commensurability, we mean weak commensurability. Therefore two discrete subgroups 1 and 2 of a Lie group G are commensurable if there exists g 2 G such that the group 1 \ g2g1 has finite index in both 1 and g2g1. We will see that quotients of a given Lie group mod two commensurable lattices, carry the same blocking property. The definition of a lattice in a Lie group requires the introduction of Haar measure. To see a brief summary of Haar measure definition and properties see Appendix C. Let µ be Haar measure on G and ⇤ ⇢ G be a discrete subgroup. Then there exists a unique (up to a scalar multiple) -finite, G-invariant Borel measure ⌫ on G/⇤ which can be defined via natural quotient map. (See Proposition 4.1.3 in [31] for proof). The Haar measure µ on G is given by a smooth volume form. so the associated measure ⌫ on G/⇤ is also given by a volume form, Therefore we say that G/⇤ has finite volume if ⌫(G/⇤) < 1. Definition 2.9. A subgroup of G is a lattice in G if • is a discrete subgroup of G, and • G/ has finite volume. Examples. If is discrete and G/ is compact, then is a lattice in G. Such a lattice is called cocompact. In addition, every finite index subgroup of a lattice is also a lattice [31, p.15, p.46]. SL(2, Z) is a lattice in SL(2, R). First, discreteness is obvious. To see the second condition we note that H2 = SL(2, R)/SO(2) (See [31, p.9]) where H2 is the hyperbolic plane as a Lie group with multiplication. Let F = {z 2 H2 | |z| 1 and 1/2  Rez  1/2} . It is well known that F is a fundamental domain for the action of SL(2, Z) on H2; it therefore suces to show that F has finite volume, or, more precisely, finite hyperbolic area. The hyperbolic area dA of an infinitesimal rectangle is the product of its hyperbolic length and its hyperbolic width. 17 If the Euclidean length is dx and the Euclidean width is dy, and the rectangle is located at the point x + iy, then, by definition of the hyperbolic metric, the hyperbolic length is (dx)/(2y) and the hyperbolic width is (dy)/(2y). Therefore, dA = dxdy 4y2 . Since Imz p3/2 for all z 2F , we have dA π 1 vol(F) =πx+iy2F p3/2π 1/2 1/2 dxdy 4y2 = 1 4π 1 p3/2 1 y2 dy < 1 . Similar but more complicated calculations show that SL(n, Z) is a lattice in SL(n, R). As in the above example, the hard part is to find a fundamental domain for the action of on G (or an appropriate approximation of a fundamental domain); then it is not dicult to see its volume is finite. These are special cases of the following general theorem which implies that every simple Lie group has a lattice (Theorem 1.3.9 in [31]). Theorem 2.10. Assume G ⇢ SL(n, R) and there exist simple Lie groups G1, · · · , Gm such that G = G1 ⇥· · ·⇥ Gm. Moreover, assume G \ SL(n, Q) is dense in G. Then GZ = G \ SL(n, Z) is a lattice in G. Lattices constructed by taking the integer points of G in this way are said to be arithmetic. When n is large, there is more than one way to embed G in SL(n, R), and dierent embeddings can lead to quite dierent intersections with SL(n, Z). In particular, for a non-compact simple Lie group G, we can take an appropriate embedding of G in some SL(n, R), and construct a non cocompact lattice in G; we also can take a dierent embedding, and construct a cocompact lattice 0 in G [31, p.16]. 2.2 Connection Blocking in Lattice Quotients of Connected Lie Groups A dierentiable manifold M with a transitive action of a Lie group G on it is called a homogeneous space of G. It can be shown that any homogeneous space of G is isomorphic to G/H, where H ⇢ G 18 is a Lie subgroup, with the canonical G-action. Homogeneous spaces are the most important and interesting objects of geometry. Let G be a connected Lie group and ⇢ G be a lattice. acts on G through left (or right) multiplication. Let ⇢ G be a lattice in G. The lattice quotient space M = G/ is a homogeneous space of G. In this section we study connection blocking properties of M = G/, following the notation and text in Gutkin [15]. For g 2 G, m 2 M, g · m denotes the action of G on M. Let g be the Lie Algebra of G and let exp : g ! G be the exponential map. For m1, m2 2 M let Cm1,m2 be the set of parametrized curves c(t) = exp(tx) · m, 0  t  1, such that c(0) = m1, c(1) = m2. We say that Cm1,m2 is the collection of connecting curves for the pair m1, m2. Let I ⇢ R be any interval. If c(t), t 2 I, is a curve, we denote by c(I)⇢ M the set {c(t) : t 2 I}. A finite set B ⇢ M \ {m1, m2} is a blocking set for the pair m1, m2 if for any curve c in Cm1,m2 we have c([0, 1]) \ B , ;. If a blocking set exists, the pair m1, m2 is connection blockable, or simply blockable. We also say that m1 is blockable (resp. not blockable) away from m2. The analogy with Riemannian security [12, 25, 21, 4] suggests the following: Definition 2.11. Let M = G/ be a lattice quotient. i) M is connection blockable if every pair of its points is blockable. If there exists at least one non-blockable pair of points in M, then M is non-blockable. ii) M is uniformly blockable if there exists N 2 N such that every pair of its points can be blocked with a set B of cardinality at most N. The smallest such N is the blocking number for M. iii) A pair m1, m2 2 M is midpoint blockable if the set {c(1/2) : c 2 Cm1,m2} is finite. A lattice quotient is midpoint blockable if all pairs of its points are midpoints blockable. iv) A lattice quotient is totally non-blockable if no pair of its points is blockable. 19 Blocking property of lattice quotients carries some straightforward and expected properties which can be summarized in the following proposition. Proposition 2.12. Let M = G/ where ⇢ G is a lattice, and let m0 = be the identity element of M. Then the following holds: i) The lattice quotient M is blockable (resp. uniformly blockable, midpoint blockable) if and only if all pairs m0, m are blockable (resp. uniformly blockable,midpoint blockable). The space M is totally non-blockable if and only if no pair m0, m is blockable; ii) Let ˜ ⇢ be lattices in G, let M = G/, ˜M = G/ ˜, and let p : ˜M ! M be the covering. ˜m1, ˜m2 2 ˜M be such that m1 = p( ˜m1), m2 = p( ˜m2). If B ⇢ M is a ˜B ⇢ ˜M is a blocking set for ˜m1, ˜m2) then p1(B) (resp. p( ˜B) Let m1, m2 2 M and let blocking set for m1, m2 (resp. is a blocking set for ˜m1, ˜m2 (resp. m1, m2). iii) Let G0, G00 be connected Lie groups with lattices 0 ⇢ G0, 00 ⇢ G00, and let M0 = G0/0, M00 = G00/00. Set G = G0⇥G00, M = M0⇥M00. Then a pair(m01, m001),(m02, m002)2 M is connection blockable if and only if both pairs m01, m02 2 M0 and m001, m002 2 M00 are connection blockable. Proof. Claim i) is immediate from the definitions. The proofs of claim ii) and claim iii) are analogous to the proof of their counterparts for riemannian security. See Proposition 1 in Gutkin [18] for claim ii), and Lemma 5.1 and Proposition 5.2 in Burns [5] for claim iii). ⇤ We say lattice quotients M1, M2 have identical blocking property if both are blockable (or not), midpoint blockable (or not), totally non-blockable (or not), etc. Recall that two subgroups 1, 2 ⇢ G are commensurable, 1 ⇠ 2, if there exists g 2 G such that the group 1 \ g2g1 has finite index in both 1 and g2g1. Commensurability yields an equivalence relation in the set of lattices in G. We will use the following immediate Corollary of Proposition 2.12. Corollary 2.13. i) If lattices 1, 2 ⇢ G are commensurable, then the lattice quotients Mi = G/i , i = 1, 2 have identical blocking properties. 20 ii) Let M1 = G1/1, M2 = G2/2 be lattice quotients. Then M1 ⇥ M2 (G1 ⇥ G2)/(1 ⇥ 2) is blockable (resp. midpoint blockable, uniformly blockable) if and only if both M1 and M2 are blockable (resp. midpoint blockable, uniformly blockable). Let exp : g ! G be the exponential map. For ⇢ G denote by p : G ! G/ the projection, and set exp = p exp : g ! G/. We will say that a pair (G, ) is of exponential type if the map exp is surjective. Let M = G/. For m 2 M set Log(m) = exp1 (m). Note, Log(m) may have more than one element. We will use the following basic fact to prove a point is not blockable from identity. Proposition 2.14. Let G be a Lie group, ⇢ G a lattice such that (G, ) is of exponential type, and let M = G/. Then m 2 M is blockable away from m0 if and only if there is a map x 7! tx of Log(m) to (0, 1) such that the set {exp(tx x) : x 2 Log(m)} is contained in a finite union of -cosets. Proof. Connecting curves are cx(t) = exp(tx)/ for some x 2 Log(m). Since c(1) = m, there is 2 such that exp(x) = g. Thus c(t) = exp(t log(g)) · m0 for some 2 , and every such curve is connecting m0 with m. (2.2.1) Suppose m is blockable away from m0, and let B ⇢ G/ be a blocking set. Let tx 2( 0, 1) be such that cx(tx)2 B. Set A = {exp(tx x) : x 2 Log(m)} ⇢ G. Then (A/)⇢ B, hence finite. Thus A is contained in a finite union of -cosets. On the other hand, if for any collection {tx 2( 0, 1) : x 2 Log(m)} the set A = {exp(tx x) : x 2 ⇤ Log(m)} is contained in a finite union of -cosets, then (A/)⇢ M is a finite blocking set. If A ⇢ G is any subset, we will say that Sqrt(A) = {g 2 G : g2 2 A} (2.2.2) is the square root of A. We will say that a pair (G, ) is of virtually exponential type if there exists ˜ ⇠ such that (G, ˜) is of exponential type. 21 Corollary 2.15. Let ⇢ G be a lattice such that (G, ) is of virtually exponential type. Then: i) The lattice quotient M = G/ is midpoint blockable if and only if the square root of any coset g is contained in a finite union of -cosets. ii) Any point in M is midpoint blockable away from itself if and only if the square root of is contained in a finite union of -cosets. Proof. By Corollary 2.13, we can assume that (G, ) is of exponential type. Set tx ⌘ 1/2 in Proposition 2.14. ⇤ The following lemma relates blocking property of a lattice quotient and its closed subspaces. The proof is straightforward and is left to the reader. Lemma 2.16. Let G be a Lie group, and let ⇢ G be a lattice. Let H ⇢ G be a closed subgroup such that \ H is a lattice in H. Let X = G/,Y = H/( \ H) be the lattice quotients, and let Y ⇢ X be the natural inclusion. i) If Y is not blockable (resp. not midpoint blockable, etc) then X is not blockable (resp. not midpoint blockable, etc). ii) If Y contains a point which is not blockable (resp. not midpoint blockable) away from itself, then no point in X is blockable (resp. not midpoint blockable) away from itself. Solvable Lie groups (resp. Lie algebras) and the semisimple Lie groups (resp. Lie algebras) form two large and generally complementary classes. Every connected Lie group G is a semidirect product of a solvable Lie group R, and a semisimple Lie group S (Theorem B.4) which is called the Levi decomposition. Connection blocking in lattice quotients of connected Lie groups satisfying Raghunathan’s condition defined in the following definition, is related to connection blocking of its components quotients. Definition 2.17. Let G be a connected Lie group. The maximum connected closed nilpotent subgroup of G is called the nilradical of G. Let G = R o S be the Levi decomposition, where 22 R = Rad G is the radical of G and S is semisimple. Let denote the action of S on R. We say G satisfies Raghunathan’s condition if the kernel of has no compact factors in its identity component. Proposition 2.18. Let G be a connected Lie group which is not semisimple and satisfies Raghu- nathan’s condition and let ⇢ G be a lattice. Let G = R o S be the Levi decomposition, and N  R be the nilradical of G. Then i) \ R and \ N are lattices in R and N, respectively. ii) If R/( \ R) is non-blockable, then G/ is non-blockable. iii) If N is not abelian, then G/ is non-blockable. Proof. See Raghunathan [34, Corollary 8.28] for the proof of i). Since R is a closed (nontrivial) subgroup of G, Lemma 2.16 implies ii). Note that N/( \ N) is a nilmanifold. If N is not abelian, by Proposition 1.6 N/( \ N) is non-blockable. Now iii) follows from Lemma 2.16. ⇤ Proposition 2.18 is particularly interesting since it relates connection blocking in lattice quotients of G to its algebraic structure, the structure of its nilradical. In addition, Proposition 2.18 implies that proving Conjecture 1.4 for lattice quotients of solvable and semisimple Lie groups, also proves the conjecture for lattice quotients of all connected Lie groups satisfying Raghunathan’s condition. 23 CHAPTER 3 CONNECTION BLOCKING IN QUOTIENTS OF SOL In this chapter we investigate blocking properties in lattice quotients of Sol, an important non- nilpotent solvable Lie group and one of the eight homogeneous Thurston 3-geometries. We prove that all lattice quotients of Sol are non-blockable. 3.1 Sol and One Parameter Subgroups In this section we derive an explicit formula for one parameter subgroups in Sol, which is essential to study its blocking properties. Definition 3.1. By Sol we mean the Lie group R2 o R where z 2 R acts on R2 as ez 0 ©≠≠´ , 0 ez™Æƨ so as multiplication is given by (x1, y1, z1)(x2, y2, z2) = (x1 + ez1 x2, y1 + ez1 y2, z1 + z2), together with a left invariant Riemannian metric ds2 = e2zdx2 + e2zdy2 + dz2. Consider the three curves R ! Sol given by 1 : t 3 : t 7! (0, 0, t). These have tangent vectors 7! (t, 0, 0), 2 : t 7! (0, t, 0) and @1 @t = @ @x , @2 @t = @ @ y , @3 @t = @ @z at (0, 0, 0), respectively, and these vectors span the tangent space at that point. The left action of the group on these vectors gives a collection of three invariant vector fields X1, X2 and X3 which form a basis for the tangent space at each point. Since (x, y, z)1 7! (x + ezt, y, z), (x, y, z)2 7! (x, y + ezt, z), and (x, y, z)3 7! (x, y, z + t), it follows that X1(x, y, z) = @ @t(x, y, z)1t=0 = ez @ @x , X2(x, y, z) = 24 @ @t(x, y, z)2t=0 = ez @ @ y , and X3(x, y, z) = @ @t(x, y, z)3t=0 = @ @z . We construct the metric to be orthogonal at every point with respect to these vector fields. Thus ( ( ( @ @ @x(x,y,z) @ y(x,y,z) @z(x,y,z) @ , , , @ @ @x(x,y,z) @ y(x,y,z) @z(x,y,z) @ ) = (ezX1(x, y, z), ezX1(x, y, z)) = e2z , ) = (ezX2(x, y, z),ezX2(x, y, z)) = e2z , ) = (X3(x, y, z), X3(x, y, z)) = 1 , and so we obtain the metric given above. Let sol denote the Lie algebra of left invariant vector fields in Sol, together with the basis X1, X2, X3 as above. We have the following proposition: Proposition 3.2. The exponential map of Sol is given as the following: Given any vector X = a1X1 + a2X2 + a3X3 2 sol, exp(tX) =✓ a1 a3(ea3t 1), a2 a3(ea3t 1), a3t◆ , if a3 , 0. If a3 = 0, exp(tX) = (a1t,a2t, 0). Proof. Let (t) = (x(t), y(t), z(t)) be the integral curve to X so that 0(t) = X(t) = a1ez @ @x a2ez @ @ y + a3 @ @z . This leads to the first order system x0(t) = a1ez(t), y0(t) = a2ez(t), z0(t) = a3, (0) = (0, 0, 0) which can be easily solved giving the exponential formula. ⇤ Remark. For every g 2 Sol, the exponential map formula shows that the equation exp(X) = g has a unique solution. Let gt = exp(t log g) be the unique one parameter subgroup joining identity and g. A direct computation gives us the following corollary. 25 Corollary 3.3. If g = (x, y, z)2 Sol and z , 0, ez 1(etz 1), gt =⇣ x If g = (x, y, 0), gt = (tx,t y, 0). y ez 1(etz 1), tz⌘ . 3.2 Lattices in Sol A complete classification of lattices in Sol is presented in Molnár [30]. In this paper, Sol lattices are classified in an algorithmic way into 17 dierent types, but infinitely many Sol ane equavalence classes, in each type. For the purpose of the blocking problem, we consider a class of lattices constructed by the following proposition. We then prove, every lattice in Sol is conjugate to a lattice in this class. Proposition 3.4. Let A 2 SL(2, Z). Suppose that A is conjugate in GL(2, R) to a matrix of the form ©≠≠´ 0 0 1™Æƨ for some positive , 1. Then there is a monomorphism Z2 oA Z ,! Sol and the image is a lattice. Note that by Z2 oA Z we mean the semidirect product where r 2 Z acts on Z2 as Ar so as the multiplication is given by (p1, q1, r1)(p2, q2, r2) = ((p1, q1) + Ar1(p2, q2), r1 + r2). Proof. By assumption there exists P 2 GL(2, R), and s 2 R \ {0} such that = es, and . PAP1 =©≠≠´ 0 0 1™Æƨ 26 Define the embedding by (p, q, r) 7! (P(p, q), sr) and note that since s , 0, and P is nonsingular this is an injection. The following calculation demonstrate that this gives a homomorphism: (p1, q1, r1)(p2, q2, r2) = ((p1, q1) + Ar1(p2, q2), r1 + r2) r1 0 7!P(p1, q1) + PAr1(p2, q2), s(r1 + r2) = (P(p1, q1) +©≠≠´ = (P(p1, q1) +©≠≠´ es™Æƨ esr1™Æƨ = (P(p1, q1), sr1)(P(p2, q2), sr2) . es 0 esr1 0 P(p2, q2), sr1 + sr2) 0 P(p2, q2), sr1 + sr2) The quotient of Sol by Z2 oA Z is a T2 bundle over S1 so is compact. Thus Z2 oA Z is indeed a lattice in Sol. We now show that the action of Z2 oA Z on Sol is proper. Let g = (X,Y, Z)2 Sol and let = (p, q, r)2 Z2 oA Z \ {1}. Then g = (P(p, q) + (er X, erY), sr + Z). If r , 0 then s| 0. If r = 0 then g = (P(p, q) + (X,Y), Z) and both g and g lie in the same d(g, g)| horizontal plane z = Z on which the metric restricts to ds2 = e2Z dx2+e2Z dy2+dz2. In this case let µ = min{e2Z, e2Z} > 0 and let K = infk(x,y)k2=1 kP(x, y)k2 > 0. Then d(g, g) µKk(p, q)k2 and since , 1, (p, q) , (0, 0) so d(g, g) µK. We have thus shown that for all 2 Z2 oA Z with , 1, d(g, g) min{s, µK} > 0. Hence the action of Z2 oA Z on Sol is proper. ⇤ Sol multiplication can be projectively interpreted by "left translations" on its points as L⌧ : (x, y, z) 7! ⌧(x, y, z),⌧ 2 Sol. Let L(T) denote the set of left translations on Sol and assume < L(T) is a subgroup, generated by three independent translations ⌧1 = (x1, y1, z1),⌧ 2 = (x2, y2, z2),⌧ 3 = (x3, y3, z3) with non-commutative addition, or in this case Z linear combinations. Notation. Let ⌧1 and ⌧2 be left translations. [⌧1,⌧ 2] := ⌧1 2 ⌧1⌧2 denotes the commutator. For matrices A, P 2 GL(2, R), AP := P1AP. 1 ⌧1 The concept of a lattice can be rephrased as a subgroup of left translations. The theorem below clarifies the algebraic structure of lattices in Sol (see [30]). 27 2 GL(2, R) satisfies: x1 x2 ©≠≠´ y1 y2™Æƨ ez3 0 AP = P1AP =©≠≠´ . 0 ez3™Æƨ Theorem 3.5. For each lattice of Sol there exists A =©≠≠´ ⌧i = (xi, yi, zi), i = 1, 2, 3 such that: i) has a group presentation a b c d™Æƨ 2 SL(2, Z) with tr(A) > 2, and = (A) = h⌧1,⌧ 2,⌧ 3 : [⌧1,⌧ 2] = 1,⌧ 1 3 ⌧1⌧3 = ⌧1AP,⌧ 1 3 ⌧2⌧3 = ⌧2APi , ii) ⌧1 = (x1, y1, z1),⌧ 2 = (x2, y2, z2) satisfy the equalities z1 = z2 = 0, and the matrix P = Remark. Molnár’s definition of Sol multiplication is slightly dierent from our definition. In his paper he defines the lattices as a subgroup of right translations of Sol. As a result, the statement of Theorem 3.5 has been readjusted accordingly. Using notations of Proposition 3.4 and Theorem 3.5, it’s easy to see that lattices of Proposition 3.4, correspond to (A) = h⌧1,⌧ 2,⌧ 3i where, x3 = y3 = 0. Then ez3, ez3 are eigenvalues of A and P is the eigenvector matrix of A. Now pairing Proposition 3.4 and Theorem 3.5, we conclude the following proposition which will be used later to study blocking property of all quotients of Sol. Proposition 3.6. Every lattice of Sol is conjugate to a semidirect product lattice presented by Proposition 3.4. Proof. Given a lattice = (A) = h⌧1,⌧ 2,⌧ 3i as in Theorem 3.5, let 0 = 0(A) = h⌧1,⌧ 2,⌧ 03 = ez31, 0⌘, g 2 Aut(Sol) : (x, y, z) 7! g1(x, y, z)g. Since g commutes (0, 0, z3)i, g =⇣ x3 ez31, y3 28 with ⌧1,⌧ 2, g(⌧1) = ⌧1, g(⌧2) = ⌧2. In addition, we compute: g(⌧03) = g1⌧03g =⇣ =⇣ x3 ez3 1, x3 ez3 1, y3 ez3 1, 0⌘ (0, 0, z3)⇣ ez3 1, z3⌘⇣ x3 ez3 1, y3 x3 ez3 1, ez3 1, 0⌘ y3 y3 ez3 1, 0⌘ = (x3, y3, z3) = ⌧3 . Hence g(0) = . ⇤ 3.3 Blocking Property of Sol Quotient Spaces This section concludes with the proof of the main theorems. We first need a few technical lemmas that will be applied in the body of the proofs. Lemma 3.7. For an integer n > 2, n2 4 is never a perfect square. Proof. By contrary suppose there exist a positive integer k such that n24 = k2, so that (n k)(n + k) = 4. Noting 0 < n k < n + k, it follows that n k = 1 and n + k = 4, and thus n = 5/2 contradicting the assumption. ⇤ 2 SL(2, Z) with eigenvalues = es , 1, 1. Then < Q. The Lemma 3.8. Let A =©≠≠´ matrix a b c d™Æƨ is invertible, and Given such a matrix P, 1 c(es d) 1 ™Æƨ 1 1 b(es a) P =©≠≠´ PAP1 =©≠≠´ 0 0 1™Æƨ . 1 b(es a)p, sr) . (P(p, q), sr) = (p 1 c(es d)q, q 29 , 1 0 1 c(es d) v2 =©≠≠´ ˜P1A ˜P =©≠≠´ PAP1 =©≠≠´ ˜P =©≠≠´ ™Æƨ 0 1™Æƨ 0 1™Æƨ 0 . . Let P = ( ˜P/det( ˜P))1. Following (3.3.1) it’s easy to see that Proof. Since + 1 = tr(A), solving the quadratic equation for it follows that = tr(A)/2 ± ptr(A)2 4/2. Note that tr(A)2 Z and tr(A) = (es + es) > 2. Now Lemma 3.7 implies tr(A)2 4 is not a perfect square, so it’s irrational. Let v1, v2 be the eigenvectors associated to , 1, so that the first component of v1 and the second component of v2 are equal to 1, respectively, and assume ˜P = [v1, v2]. A direct computation shows that: , v1 =©≠≠´ 1 1 b(es a)™Æƨ ˜P is the eigenvector matrix, so it’s invertible and 1 1 b(es a) 1 c(es d) 1 ™Æƨ . (3.3.1) Using the common formula to find inverse of the (2⇥2) matrix ˜P/det( ˜P) and noting det( ˜P/det( ˜P)) = (det( ˜P))1, if follows that: P = 1 det( ˜P)1©≠≠´ det( ˜P)1 det( ˜P)1 1 b(es a) det( ˜P)1 1 c(es d) det( ˜P)1 ™Æƨ . ⇤ The last statement of the Lemma follows from direct computation. Next we prove the following lemma. Lemma 3.9. Let A be conjugate to its eigenvalue matrix via matrices P1, P2 2 GL(2, R) as in the images statement of Proposition 3.4, and i, i = 1, 2 be the two associated lattices in Sol, i.e. of the embeddings (p, q, r) 7! (Pi(p, q), sr). Then B = P2P1 is diagonal and the mapping 1 : Sol ! Sol, (x, y, z) 7! (B(x, y), z) is a Lie group automorphism. In addition (1) = 2, hence quotient spaces Sol/1 and Sol/2 have identical blocking property, i.e. m = g1 is blockable from the identity m0 1 = 1 if and only if (g)2 is blockable from m0 2 = 2. 30 1 , P1 2 1 = P1 Proof. Since both P1 are eigenvector matrices of A, there exists a diagonal and invertible 2 B, and thus B = P2P1 matrix B such that P1 1 . Since B is nonsingular, is a dieomorphism on Sol, and it’s clear from the definition (1) = 2. The following calculation demonstrates that the mapping is a homomorphism: ez1 0 (x1, y1, z1)(x2, y2, z2) = ((x1, y1) +©≠≠´ 7! (B(x1, y1) + B©≠≠´ = (B(x1, y1) +©≠≠´ ez1 0 ez1 0 0 0 ez1™Æƨ ez1™Æƨ ez1™Æƨ 0 = (B(x1, y1), z1)(B(x2, y2), z2) . (x2, y2), z1 + z2) (x2, y2), z1 + z2) B(x2, y2), z1 + z2) Since Lie group isomorphisms map one parameter subgroups to one parameter subgroups, the last statement follows immediately. ⇤ Now we are ready to prove the main theorems. 2 SL(2, Z) with eigenvalues , 1, 1 and P11 = P22 = 1 (Note that since a b Proof of Theorem 1.7. Assume that matrix A =©≠≠´ c d™Æƨ is given, and P 2 GL(2, R) is such that PAP1 =©≠≠´ 0 1™Æƨ 0 switching A $ A doesn’t change P we may assume > 0 and since , 1, tr(A) > 2). Let = es, s , 0, be the image lattice of Z2 oA Z through the embedding in Proposition 3.4, g = (0, y, z)2 Sol, y , 0, and m = g. We prove that m is not blockable from the identity m0 = . Changing the representative g for m = g if necessary, we may assume z , 0. To the contrary, assume that m is blockable from identity m0. Let ri be a sequence of integers, so that sri is strictly increasing and, sri ! 1, as i ! 1, and let i = (0, 0, sri)2 . By Proposition 2.14, for a suitable choice of ti’s where 0 < ti < 1, there exist ˜g1, · · · , ˜gn 2 Sol such that {(gi)ti}⇢ [ N n=1 ˜gn; passing 31 to a subsequence if necessary, we may assume there exists a fixed ˜g 2 Sol such that for each i, (gi)ti 2 ˜g. In addition we may assume ˜g = (g1)t1. In particular, there exist ˜y, ˜z 2 R such that ˜g = (g1)t1 = (0, ˜y, ˜z). Since gi = (0, y, z + sri), Corollary 3.3 implies y (3.3.2) (gi)ti =⇣0, ezsri 1(eti(z+sri) 1), ti(z + sri)⌘ . [(gj)tj]2 , there exist ˜p, ˜q, ˜r 2 Z so that [(gi)ti]1 · [(gj)tj] = ( ˜p 1 a) ˜p, s˜r). Hence ˜p 1 follows that In particular, for each i, j the first component of [(gi)ti]1 · [(gj)tj] is 0. As [(gi)ti]1 · b(es c(es d) ˜q = 0, and since es = < Q, we must have ˜p = ˜q = 0. Therefore it c(es d) ˜q, ˜q 1 [(gi)ti]1 · [(gj)tj] = (0, 0, s ˜ri j), ˜ri j 2 Z, letting i = 1, we conclude that (gj)tj = (g1)t1(0, 0, s ˜rj) = ˜g(0, 0, s ˜rj) = (0, ˜y, ˜z + s ˜rj) (3.3.3) (3.3.4) which means {(gi)ti}\ ˜g lies on a vertical line in y-z plane. In addition, comparing the third component of both sides in equation 3.3.3 implies tj(z + srj) ti(z + sri) = s ˜ri j, ri j 2 Z . Solving for ti using the second components of equations 3.3.2 and 3.3.4 it follows that ti = 1 z + sri ln✓ ˜y y(ezsri 1) + 1◆ ; (3.3.5) (3.3.6) Since for each i, 0 < ti < 1, letting i ! 1 shows that ˜y/y has to be positive. Now plugging the formula for ti and tj in equation 3.3.5 gives us: ezsri 1 + y/ ˜y ezsrj 1 + y/ ˜y = es ˜ri j (3.3.7) Setting, j = i + 1, i ! 1, the left side of the above equation goes to 1. So, for large enough i, j = i + 1, ˜ri j = 0, and so ri = ri+1 which is a contradiction. 32 Now, if g = (x, 0, z), x , 0, replace (0, 0, sri) with (0, 0,sri); repeating a similar argument on ⇤ the first component of (gi)ti, proves that m = (x, 0, z) is also not blockable from m0. Knowing all lattices of Sol are conjugate to semidirect products, we are ready to prove the second theorem. Proof of Theorem 1.8. By Proposition 3.6 and Lemma 3.9 it suces to prove the theorem for a lattice presented in Theorem 1.7. From Theorem 1.7, all cosets in X = {(0, y, z) | y, z 2 R, y , 0} are non-blockable from the identity. We show that the group elements in X is dense in Sol, which implies X is dense in Sol/. Fix g = (x0, y0, z0)2 Sol, and assume ✏> 0 is given. Since es is not rational, {(ez0(p 1 c(es d)q) | p, q 2 Z} is dense in R. Let p1, q1 be such that |(ez0(p1 1 c(es d)q1) x0| <✏ /2 , moreover choose a non-zero real number y1 such that |y1 + ez0(q1 1 b(es a)p1) y0| <✏ /2 . Let g1 = (0, y1, z0)2 Sol and 1 = (P(p1, q1), 0)2 . Then 1 c(es d)q1), y1 + ez0(q1 g11 = ((ez0(p1 1 b(es a)p1), z0)2 X , and the above argument shows the Euclidean distance d(g11, g) <✏ , and hence X is dense in Sol. Thus X is a dense subset of Sol/, not blockable from the identity m0, which implies the statement of Theorem 1.8. ⇤ 33 CHAPTER 4 CONNECTION BLOCKING IN SEMISIMPLE LATTICE QUOTIENTS In this chapter we study connection blocking in lattice quotients of SL(n, R). We start with quotients of SL(2, R), and then extend the result to quotients of SL(n, R), n > 2. Since by Corollary 2.13 direct product of lattice quotients carries the same blocking property as its components, the results of the current chapter can be extended to lattice quotients of a large class of semisimple Lie groups. 4.1 One Parameter Families of SL(2, R) and Modified Times In this section we derive an explicit formula for one parameter families in SL(2, R), which is essential to study its blocking properties. The exponential map for SL(2, R) can be formulated in terms of trigonometric functions. The formula is directly derived from the exponential power series exp(X) =Õ1k=0 X k/k!, doing some matrix algebra. For details see Rossmann [35, pp. 17-19]. We have the following proposition: Proposition 4.1. Let g0 denote the identity element of SL(2, R). For a given matrix if a2 +b > 0, then let !(X) = pa2 + bc > 0 and if a2 +b < 0, then let !(X) =p(a2 + bc) > 0. In the first case we have and in the second case (a2 + bc < 0), we have a b 2 sl(2, R), c a™Æƨ X =©≠≠´ exp(X) = (cosh !) g0 +✓sinh ! ! ◆ X, exp(X) = (cos !) g0 +✓sin ! ! ◆ X. (4.1.1) (4.1.2) (4.1.3) If a2 + bc = 0, then exp(X) = g0 + X. Furthermore, every matrix g 2 SL(2, R) whose trace satisfies tr(g) 2 is in the image of the exponential map. Consequently, for any g 2 SL(2, R), either g or 34 g is of the form exp(X), for some X 2 sl(2, R). Therefore, (SL(2, R), SL(2, Z)) is of exponential type. For g 2 SL(2, R) with tr(g) 2, log(g) is unique and we use the notations !g = !(log(g)), gt = exp(t log g), 0  t  1. We have the following lemma: Lemma 4.2. If tr(g) 2, we have gt =✓cosh(t!g) sinh t!g sinh !g cosh !g◆ g0 + sinh t!g sinh !g g, where g0 is the identity element of SL(2, R). Proof. From (4.1.2) it follows that log g = !g sinh !gg cosh(!g)g0 Noting that !(t log(g)) = t!(log(g)), substituting (4.1.5) in the equation gt = exp(t log g) = cosh(t!g)g0 + sinh t!g t!g (t log g) gives the desired formula. Definition 4.3. For a fixed g 2 SL(2, R) and an arbitrary 2 . We use the notation sinh(t!g) sinh(!g) We call the modified time associated with . = , 0   1 . (4.1.4) (4.1.5) ⇤ Let a() =⇣1 +⇣tr(g)2/4 1⌘ 2 ⌘1/2 . (4.1.6) From (4.1.2) we have cosh ! = tr(g)/2; a direct computation from (4.1.5) gives the following formula 35 (g)t =⇥a() 1/2tr(g)⇤ g0 + g. Modified time as defined in above, will be pivotal for the proof of the main theorem. (4.1.7) Notation. While working with a sequence {i}2 , by i, a(i) we mean i, a(i). 4.2 Blocking Properties of Mn This section concludes with the proof of Theorem 1.9. The proof will be based on the technical Proposition 4.8, which is the main body of this section. Throughout the section, = SL(2, Z), M2 = SL(2, R)/. We assume: yi xi x g =©≠≠´ y z w™Æƨ 2 SL(2, Q), {i}⇢ , gi =©≠≠´ zi wi™Æƨ . Moreover, since g and g have identical blocking properties, we may assume x > 0. In order to prove Proposition 4.8, we first need a few Lemmas. Lemma 4.4. Suppose that R(↵, )2 R[↵, ] has the form R(↵, ) = c↵n + P(↵, ) for some n > 0, c , 0, and polynomial P(↵, ) of degree of at most n 1 in ↵. Then given any sequence of positive real numbers {i} such that i ! 1, as i ! 1, there exists an increasing function f : Z+ ! Z+ such that R( f(i), f(j)) , 0, 8i > j. Proof. Define f : Z+ ! Z+ inductively as follows. Set f(1) = 1, and assuming f(k) is defined, define f(k + 1) in the following way. The k-polynomials R1(↵) = R(↵, f(1)), · · · , Rk(↵) = R(↵, f(k)) are all degree n in ↵. Choose l large enough so that R1(l), · · · , Rk(l) , 0, and define f(k + 1) = l. ⇤ Lemma 4.5. For a given element g 2 SL(2, R): i) Every five elements of coset g are Z linearly dependent. 36 ii) Let g1, · · · , gn, n  4, be Z(or Q)-linearly independent elements of g. Then there exists a non-zero integer m0 such that for every g 2 spanQ < g1, · · · , gn >, there exists (m1, · · · , mn)2 Zn so thatÕn i=1 mi(gi) = m0(g). Proof. To prove the first part note that g 2 spanQ < g1, · · · , gn > if and only if 2 spanQ < 1, · · · , n >; therefore we may assume g = . Considering as a subset of Q-vector space Q4, immediately implies every five elements of it are Q (and therefore Z)-linearly dependent. Now we prove part ii) of the Lemma. First a conventional notation; For 2 , =©≠≠´ 1 2 3 4™Æƨ define [] = (1, 2, 3, 4)T; moreover we use the notation [a] = (a1, · · · , an)T, to denote an arbitrary element of Rn as a n ⇥ 1 matrix. Let A = ([1]· · ·[n]). Note that A is a 4 ⇥ n matrix of rank n, thus there exists an invertible n ⇥ n submatrix ˜A consisting of rows, say, i1, · · · , in. Take an arbitrary element 2 spanQ < 1, · · · , n >. Since ˜A1 has rational entries, we can choose a fixed integer m0 so that m0 ˜A1 has integer entries. Hence the linear equation ˜A[m] = m0(i1, · · · , in)T has a solution [m] = (m1, · · · , mn)T 2 Zn. For 1  j  4, j , i1, · · · , in, let Aj denote the j-th row of A, and assume Aj =Õn k=1 ↵jk Aik . Since 2 spanQ < 1, · · · , n >, there exists [r]2 Qn such that [] = A[r]. It follows that: j = Aj[r] = ( n’k=1 Hence we have: ↵jk Aik)[r] = ↵jk(Aik[r]) = ↵jk ik . n’k=1 n’k=1 ↵jkm0ik = m0 j . ⇤ Lemma 4.6. Every coset m 2 SL(2, Q)/ has a representative of the form n’k=1 n’k=1 Aj[m] = ( ↵jk Aik)[m] = ↵jk(Aik[m]) = n’k=1 Therefore we conclude A[m] = m0[], that implies m0 =Õn z 1/x™Æƨ g =©≠≠´ x 0 ; i=1 mii. 37 that is m = g, where x, z 2 Q. Proof. Let be an arbitrary representative of coset m. If y , 0, let s/q = x/y, gcd(s, q) = 1 and choose p, r 2 Z so that ps rq = 1, and let It is clear that g = g1 2 m and g has the desired form. Lemma 4.7. Let g 2 SL(2, R), and 1, · · · , n 2 have forms . r y x p q z w™Æƨ s™Æƨ g1 =©≠≠´ =©≠≠´ , x, z 2 R and i =©≠≠´ pi ri . 1 si™Æƨ Let (g1)t1, · · · ,(gn)tn be Z-linearly dependent, that is mi(gi)ti = 0, mi 2 Z . Then we haveÕn Proof. By (4.1.7) we have i=1 mia(i) = 0. x 0 g =©≠≠´ z 1/x™Æƨ n’i=1 i=1 mii = 0 andÕn (gi)ti =©≠≠´ Now (4.2.1) implies a(i) + 1/2i(xi wi) izi i yi a(i) + 1/2i(wi xi)™Æƨ . n’i=1 mii yi = 0 , n’i=1 mia(i) + 1/2i(xi wi) = 0 , 38 ⇤ (4.2.1) (4.2.2) (4.2.3) n’i=1 mia(i) + 1/2i(wi xi) = 0 . 0, add (4.2.3) and (4.2.4). Since y1 = · · · = yn = x, (4.2.2) immediately impliesÕn Proposition 4.8. Let g =©≠≠´ z 1/x™Æƨ Then there exists a sequence 0 x i=1 mii = 0. To obtainÕn (4.2.4) i=1 mia(i) = ⇤ 2 SL(2, Q), and m = g be finitely blockable from identity m0. and a sequence of times {ti}⇢( 0, 1) such that i =©≠≠´ 1 pi pisi 1 si™Æƨ 2 i) all elements of {(gi)ti} belong to the same coset, and all modified times are the same, i.e., i = = const, ii) 2 i , ia(i)2 Q, iii) Ci = tr(gi) is an increasing sequence of positive rational numbers with the same denomi- nator, Ci ! 1, as i ! 1, and iv) {pi} is an increasing sequence of positive integers. Proof. Let m = g, g 2 SL(2, Q), be blockable from identity m0. Suppose x = a/b, a, b 2 Z+. Let pi = 2ib2, si = (a 2a2)i, then a direct computation shows Ci = tr(gi) = z + ib. It is clear that passing to a subsequence if necessary, we may assume 2 < C1 < C2 < · · · . Note that Ci’s are rational numbers with the same denominator. By Proposition 2.14 for a suitable choice of ti’s where 0 < ti < 1 we have {(gi)ti}⇢ [ N n=1 ˜gn; passing to a subsequence if necessary it follows that there exists a sequence , i 2 i =©≠≠´ 1 pi pisi 1 si™Æƨ 39 such that tr(gi) = Ci = z + nib, pi = 2nib2 where ni 2 Z+, n1 < n2 < · · · and (gi)ti 2 ˜g for some fixed ˜g 2 G. Now let i = be modified times i 2( 0, 1). We show that for every pair of indexes (i, j), ij 2 Q. By (4.1.7) we have a(i) + 1/2i(xi wi) sinh(ti!i) sinh(!i) i yi , izi a(i) + 1/2i(wi xi) (gi)ti =©≠≠´ a(i) + 1/2i(wi xi)™Æƨ a(i) = a(i) =h1 +⇣1/4tr(gi)2 1⌘ 2 ii1/2 ·h(gj)tji 2 it follows that ©≠≠´ a(i) + 1/2i(xi wi)™Æƨ a(j) + 1/2j(wj xj)™Æƨ ™ÆÆÆÆÆÆÆƨ B(i, j)©≠≠≠≠≠≠≠≠´ a(i)a(j) ia(j) a(i)j ij a(j) + 1/2j(xj wj) 2 Z4 izi ©≠≠´ j z j j yj i yi · . 2 (4.2.5) where Since⇥(gi)ti⇤1 which can be written as where B(i, j) =©≠≠≠≠≠≠≠≠´ 1 1/2(wi xi) 1/2(xj wj) 0 0 1 1/2(xi wi) 1/2(wj xj) yi zi yj z j 1/4(wi xi)(xj wj) yiz j 1/2yj(wi xi) 1/2yi(wj xj) 1/2z j(xi wi) 1/2zi(xj wj) 1/4(xi wi)(wj xj) zi yj ™ÆÆÆÆÆÆÆƨ We claim that passing to a subsequence of {(gi)ti} if necessary, we may assume det(B(i, j)) , 0. Let ui = xi wi, then a direct but lengthy computation shows that det(B(i, j)) = x⇣u2 i z j + u2 j zi uiuj(zi + z j) x(z j zi)2⌘ 40 Noting that ui = 2xpi Ci = (4ab b)ni z, zi = xp2 we see that det(B(i, j)) = a2b4(2 4a)2n4 Now Lemma 4.4 proves the claim. i +Cipi 1/x = b3(24a)n2 i +2zb2ni b/a, j + P(ni, nj) where P is a third degree polynomial in nj. Now, from (4.2.5) 2 i , ij, ia(j)2 Q. Let 1  n0  4 be the biggest integer such that there are n0 Q (or Z)-linearly independent elements of (gi)ti 2 ˜g. Then it is clear that for all i, (gi)ti 2 spanQ < (g1)t1, · · · ,(gn0)tn0 >. Lemma 4.7 implies that for every i, there exist integers mi,0, mi,1 · · · , mi,n0 such that mi,0i = mi,11 + · · · + mi,n0n0 and since (gi)ti,(g1)t1, · · · ,(gn0)tn0 all belong to the same coset, by Lemma 4.5 we can assume mi,0 = m0 is nonzero and fixed. Now, from previous step and the equation 02 m2 i = (mi,11 + · · · + mi,n0n0)2 we conclude {i} does not have any accumulation point and since {i}⇢( 0, 1) it follows that it’s finite. Passing to a subsequence again, we may assume i = = const. ⇤ Now we are ready to prove Theorem 1.9. Proof of Theorem 1.9. By contrary suppose m = g 2 SL(2, Q)/ is blockable from identity m0. By Lemma 4.6 we may assume: x 0 , x, z 2 Q . g =©≠≠´ z 1/x™Æƨ Let {(gi)ti} be a sequence as in Proposition 4.8, and suppose tr(gi) = Ci = xi/y, xi, y 2 Z+, and i = 2 = k/l < 1, k, l 2 Z+. Substituting theses into (4.1.6) it follows that 2 4y2l2⇣4kl y2 4k2y2 + k2x2 i⌘ . By Proposition 4.8, ii), we have ia(i) = a(i)2 Q, so(ia(i))2 = a2 i /b2 Thus there exists ˜ai 2 Z+ so that i , for some ai, bi 2 Z+. (ia(i))2 = 1 ⇣4kl 4k2⌘ y2 + k2x2 i = ˜a2 i , 41 which can be rewritten as ⇣4kl 4k2⌘ y2 = ( ˜ai + k xi)( ˜ai k xi) . Since k < l, left side is a constant positive integer. Letting xi ! 1, the above equation yields a contradiction. ⇤ From above theorem it immediately follows: Corollary 4.9. Two elements m1 = g1 and m2 = g2 2 M2 are not blockable from each other if g1 1 g2 2 SL(2, Q), therefore the set of non-blackable pairs is a dense subset of M2 ⇥ M2. Following the proof of Proposition 9 in Gutkin [15], we prove Theorem 1.10: Proof of Theorem 1.10. For 1  i  n 1 let Gi ⇢ SL(n, R) be the group SL(2, R) embed- ded in SL(n, R) via the rows and columns i, i + 1. Then Gi \ SL(n, Z) SL(2, Z), and hence GiSL(n, Z)/SL(n, Z) SL(2, R)/SL(2, Z). Set M(i)n = GiSL(n, Z)/SL(n, Z)⇢ Mn. By Theorem 1.9, each M(i)n has infinitely many non-blockable pairs m1, m2, yielding the the claim. ⇤ 4.3 Blocking Property and Cocompact Lattices of SL(2, R) In the previous section we dealt with non-cocompact lattice qutients of SL(2, R). As stated in Corollary 1.11, all non-cocompact quotients of SL(2, R) are non-blockable. To address the cocomapct lattices, we need to know more about the structure of these lattices. There are several ways to construct cocompact lattices of SL(2, R). In this section we study blocking properties for a class of cocompact lattices, in SL(2, R), derived from quaternion algebras. We follow the notation and discussion used in Morris [31, p.118]. First we need a few preliminaries. Definition 4.10. 1. For any field F, and any nonzero a, b 2 F, the corresponding quaternion algebra over F is the ring Ha,b F = {x + yi + z j + wk | x, y, z, w 2 F}, 42 where a) addition is defined in the obvious way, and b) multiplication is determined by the relations i2 = a, j2 = b, i j = k = ji, together with the requirement that every element of F is in the center of Ha,b that k2 = k · k = (ji)(i j) = ab.) F . (Note 2. The reduced norm of g = x + yi + z j + wk 2 Ha,b F is Nred(g) = gg = x2 ay2 bz2 + abw2 2 F, where g = x yi z j wk is the conjugate of g. (Note that gh = gh.) There are a few straightforward facts left to the reader to verify, for example: Ha2,b F for any nonzero a, b 2 F, Ha,b C Mat2⇥2(C). We need the following proposition: Mat2⇥2(F) Proposition 4.11. Fix positive integers a and b, and let G = SL(1, Ha,b R ) = {g 2 Ha,b R | Nred(g) = 1}. Then: i) G SL(2, R), ii) GZ = SL(1, Ha,b iii) the following are equivalent: Z ) is an arithmetic subgroup of G, and a) GZ is cocompact in G. b) (0, 0, 0, 0) is the only integer solution (p, q, r, s) of the Diophantine equation w2 ax2 by2 + abz2 = 0. 43 c) Every nonzero element of Ha,b Q has a multiplicative inverse (so Ha,b Q is a "division algebra"). Remark. It is well known that the Diophantine equation w2 ax2 by2 + abz2 = 0 has only trivial integer solution if and only if the equation ax2 + by2 = z2 has only trivial integer solution [31, p.121]. This can happen if a, b are prime, or if a is not a square mod b, and b is not a square mod a. Throughout the section we assume a and b are such integers, so the norm equation has only trivial solution (and thus GZ is cocompact). In particular, a and b can not be perfect squares. We refer the reader to [31, p.119] for a proof. We will use the fact that the isomorphism in i) is given by: . (4.3.1) (x + yi + z j + wk) =©≠≠´ x + ypa b(z wpa) z + wpa x ypa™Æƨ Next, we discuss the exponential mapping. Let g T1(SL(1, Ha,b R )) and sl(2, R) TIdSL(2, R) be the lie algebras of G and SL(2, R) respectively. Since in equation (4.3.1) is an isomor- phism of lie groups, d1 : T1(SL(1, Ha,b R )) ! TIdSL(2, R) is a Lie algebra isomorphism. More- R ) and SL(2, R) are embedded manifolds in R4, d1 is the restriction of over, since SL(1, Ha,b the corresponding dierential when is regarded as a function from R4 to R4. Note that T1(SL(1, Ha,b R )) = {(0, u1, u2, u2) | u1, u2, u3 2 R}, computing d1 it follows that: ©≠≠≠≠≠≠≠≠´ Since the diagram 1 pa 0 0 0 0 1 pa 0 0 0 pa 1 b bpa 0 u1pa u2 + u3pa bu2 bpau3 u1pa ™ÆÆÆÆÆÆÆƨ (4.3.2) (4.3.3) 0 u1 u2 u3 ™ÆÆÆÆÆÆÆƨ ·©≠≠≠≠≠≠≠≠´ ™ÆÆÆÆÆÆÆƨ =©≠≠≠≠≠≠≠≠´ sl(2, R) exp SL(2, R) d1 44 g exp G commutes Proposition 4.1 easily implies the following: Proposition 4.12. Let G = SL(1, Ha,b let ! =q|u2 1a + u2 R ) and g R3 be its Lie algebra. Given U = (u1, u2, u3)2 g, u1i + ! 1a + u2 2b u2 3ab > 0, u3k, if u2 3ab|. Then we have the following: sinh ! sinh ! 2b u2 i) exp(U) = cosh ! + ii) exp(U) = 1 + u1i + u2 j + u3k, if u2 sin ! iii) exp(U) = cos ! + ! For g = x + yi + z j + wk 2 G with x > 1, log(g) is unique; let !g = !(log(g)), gt = exp(t log g), 0  t  1. The following Lemma is the counterpart to Lemma 4.2 and is stated as follows: ! 3ab = 0, and u3k, if u2 u2 j + 2b u2 sin ! ! ! 1a + u2 1a + u2 2b u2 sin ! ! u1i + 3ab < 0. sinh ! u2 j + Lemma 4.13. Let g = x + yi + z j + wk 2 G with x > 1, we have: gt =✓cosh(t!g) sinh t!g sinh !g cosh !g◆ 1 + sinh t!g sinh !g g. Proof. Follow the steps of Lemma 4.2. (4.3.4) ⇤ Let = SL(1, Ha,b an arbitrary 2 , = can easily conclude: Z ) be a cocompact lattice. Following notations of Section 2, for a fixed g and , 0   1 is the modified time. Through similar step we sinh(t!g) sinh(!g) where (g)t =⇥a() x⇤ 1 + g, a() =⇣1 +⇣x2 1⌘ 2 ⌘1/2 . (4.3.5) (4.3.6) To follow through the proof of Proposition 4.8 for cocompact lattices we only consider elements g = x + yi 2 SL(1, Ha,b Q ). For a sequence {i}⇢ let gi = xi + yii + zi j + wik. We need the following lemma. 45 1 as2 1 , 0. Let n = p2 1 aq2 1. i aw2 Lemma 4.14. Let g = x + yi 2 SL(1, Ha,b such that zi and wi in gi, are nonzero and fixed for all i, z2 Q ). There exists a sequence i = pi + qii + ri j + sik 2 , i , 0, and xi ! 1, as i ! 1. Proof. Fix an element 1 = p1 + q1i + r1 j + s1k 2 such that xr1 + ays1 , 0, and xs1 + yr1 , 0. Since a is not a perfect square, r2 It is well known that if the Pell’s equation p2 aq2 = n has one solution (and a is not a perfect square), it has infinitely many solutions. Let (pi, qi)2 Z2 be an infinite set of distinct solutions such that xpi, yqi > 0, and let i = pi + qii + r1 j + s1k. Then it is easily seen zi = xr1 + ays1, wi = xs1 + yr1 are fixed, z2 i aw2 ⇤ It can be easily seen Lemma 4.5 is valid for the cocompact lattices , if we think of elements of as two by two matrices with integer entries. The following proposition is the counterpart to Proposition 4.8 for cocompact lattices. 1) , 0, and xi = xpi + ayqi ! 1 as i ! 1. i = (x2 ay2)(r2 1 as2 Proposition 4.15. Let g = x + yi 2 SL(1, Ha,b Q ). m = g, is finitely blockable from identity m0, then there exists a sequence i = pi + qii + ri j + sik and a sequence of times {ti}⇢( 0, 1) such that i) all elements of {(gi)ti} belong to the same coset, and all modified times are the same, i.e., i = = const, ii) 2 i , ia(i)2 Q, iii) xi = Re(gi) is an increasing sequence of positive rational numbers with the same denomi- nator, xi ! 1, as i ! 1, and iv) {pi} is an increasing sequence of positive integers. Proof. Let m = g, be blockable from identity m0. Let {i} be a sequence as in Lemma 4.14. Then xi = Re(xi) is an increasing sequence of rational numbers with the same denominator, and xi ! 1, as i ! 1. By Proposition 2.14 for a suitable choice of ti’s where 0 < ti < 1 we should have {(gi)ti}⇢ [ N n=1 ˜gn; passing to a subsequence if necessary, we may assume (gi)ti 2 ˜g for some fixed ˜g 2 G. 46 sinh(ti!i) sinh(!i) be modified times i 2( 0, 1). We show that for every pair of indexes Now let i = (i, j), ij 2 Q. By (4.3.5) and (4.3.6) we have (gi)ti = a(i) + i(yii + zi j + wik) where a(i) =⇣1 +⇣x2 1⌘ 2 i⌘1/2 . Since⇥(gi)ti⇤1 ·h(gj)tji 2 it follows that (a(i) i(yii + zi j + wik)) ·⇣a(j) + j(yji + z j j + wj k)⌘ 2 which can be written as where B(i, j)©≠≠≠≠≠≠≠≠´ a(i)a(j) ia(j) a(i)j ij ™ÆÆÆÆÆÆÆƨ 2 Z4 (4.3.7) B(i, j) =©≠≠≠≠≠≠≠≠´ 0 wiwjab yi yja ziz jb 0 1 0 yi yj 0 zi z j 0 wi wj (ziwj wiz j)b (wi yj yiwj)a zi yj yiz j ™ÆÆÆÆÆÆÆƨ We claim that passing to a subsequence of {g ti i } if necessary, we may assume det(B(i, j)) , 0. A direct computation shows that det(B(i, j)) = (wi yj yiwj)2a ( wiz j ziwj)2b + (zi yj yiz j)2 Note that det(B(i, j)) is a second degree polynomial in yj (the coecient of y2 i aw2 so by Lemma 4.4 and passing to a subsequence if necessary, we may assume det(B(i, j)) , 0. j is z2 i , 0); 47 Now, from (4.3.7) 2 i , ij, ia(j)2 Q. Let 1  n0  4 be the biggest integer such that there are n0 Q (or Z)-linearly independent elements of (gi)ti 2 ˜g. Then it is clear that (gi)ti 2 spanQ < (g1)t1, · · · ,(gn0)tn0 >, for arbitrary i which implies, considering the z-component, mi,0izi = mi,11z1 + · · · + mi,n0n0zn0 . By Lemma 4.14 zi = z is nonzero and fixed, and since (gi)ti,(g1)t1, · · · ,(gn0)tn0 all belong to the same coset, by Lemma 4.5 we can assume mi,0 = m0 , 0 is also fixed and does not depend on i. Now, from previous step and the equation 02 m2 i = (mi,11 + · · · + mi,n0n0)2 we conclude {i} does not have any accumulation point and since {i}⇢( 0, 1) it follows that it’s finite. Passing to a subsequence again, we may assume i = = const. ⇤ Proof of Theorem 1.12. The proof is quite similar to proof of Theorem 1.9, just replace Ci with 2xi, and Proposition 4.8 with Proposition 4.15. ⇤ 48 CHAPTER 5 CONCLUDING REMARKS AND PROPOSED PROBLEMS FOR FURTHER RESEARCH 5.1 Connection Blocking Problems in Other Lattice Quotients In the context of solvable Lie groups, one can study the blocking problem in higher dimensional versions of Sol. Let A be a positive definite (symmetric) n ⇥ n matrix with no eigenvalues equal to one. By SolA := Rn1 oA R we mean semi direct product of Rn1 and R, where t 2 R acts on Rn1 as At, i.e. (x1, t1)(x2, t2) = (x1 + At1x2, t1 + t2), x1, x2 2 Rn1, t1, t2 2 R. SolA is an n-dimensional solvable Lie group which is a generalization of Sol. An argument similar to the proof of Proposition 3.4 shows that there is a monomorphism Zn1 oA Z ,! SolA and the image ⇢ SolA is a lattice. It would be then natural to ask the following questions: Q1: How other lattices in SolA are related to ? Q2: Are all lattice quotients of Sol(n) non-blockable? Delving into the method of proof for non-blockability of Sol quotients reveals that the method might be applicable to SolA/. It would also be interesting to investigate the connection between SolA and n 1-dimensional hyperbolic space Hn1. In the context of lattice quotients of semisimple Lie groups there is a lot of room to work on the connection blocking problem. By Margulis Arithmeticity Theorem (see Theorem D.5), every lattice of SL(n, R), n 3 is arithmetic, that is its algebraic structure looks very similar to SL(n, Z). In fact modding out normal compact subgroups, arithmetic subgroups are commensurable to an integer points lattice, i.e. a lattice of the form GZ = G \ SL(n, Z), where G is a subgroup of SL(n, R) (see appendix D). Since connection blocking is invariant through modding out normal compact subgroups and commensurability, it suces to consider integer points lattice of the form GZ. It would be then worthwhile to investigate the possibility of applying modified times method 49 to a lattice GZ and prove the following quotient spaces are non-blockable. The other interesting problem is studying finite blocking for quotients of Special orthogonal group, SO(m, n) := {g 2 SL(m + n, R)|gT Im,ng = Im,n} (Im,n = diag(1, 1, · · · , 1,1,1, · · · ,1)2 Mat(m+n)⇥(m+n)(R), where the number of 1’s is m and the number of -1’s is n). This paves the way to prove lattice quotients of every linear, semisimple Lie group G are non-blockable. Indeed, without losing any main ideas, it may be assumed that G is either SL(n, R) or SO(m, n), or a prodcut of these, Morris [31, p.43]. The security problem in Riemannian manifolds and connection blocking in lattice quotient are closely related problems. Given a connected Lie group G with a left invariant Riemannian metric dg, in general one parameter subgroups passing through the identity are not geodesics. However if certain conditions are met these two classes of paths coincide. In particular, if G admits a bi-invariant Riemannian metric, the one parameter subgroups are also geodesics. A Lie group G admits a bi-invariant metric if the adjoint group Ad(G) := {Ad(g) | g 2 G} is relatively compact, i.e. it is included in a compact set (See Theorem 2, Pennec [33]). For compact Lie groups, the adjoint group is the image of a compact set by a continuous mapping and is thus also compact. Thus, bi-invariant metrics exist in such a case. Let G be a connected Lie group with a bi-invariant Riemannian metric dg, and let M = G/ be a lattice quotient of G. Then dg induces a Riemannian metric on the lattice quotient M through the projection map. In addition, geodesics passing through the identity and one parameter subgroups of M coincide. Thus for such a lattice quotient, investigating security of M as a Riemannian manifold and its connection blocking property are the same problem. 5.2 Behavior of Exponential Map Near Singularities Security of a closed Riemannian manifold has been verified for various classes of manifolds without conjugate points. Schmidt and Lafont have shown secure compact non-positively curved Riemannian manifolds are flat, Lafont [25]. Burns and Gutkin [5] prove that compact Riemannian 50 manifolds with no conjugate points and positive topological entropy are totally insecure. They also prove that uniformly secure closed Riemannian manifolds without conjugate points are flat. Both of these result deal with manifolds without conjugate points. It seems for dealing with manifolds with conjugate points, understanding the behavior of the exponential map near singularities is essential. A configuration in M is an ordered pair of points in M. Let (x, y) be a configuration in M. A geodesic joins x to y, if x is its initial point and y is its final point. A geodesic connects x and y if it joins x and y and does not pass through either x or y. Let GT(x, y) and T(x, y) denote the set of geodesics joining, and connecting x and y with length  T, respectively; nT(x, y) := |GT(x, y)| and mT(x, y) := |T(x, y)|. Moreover, let sT(x, y) be the minimal cardinality of a blocking set for T(x, y). Burns and Gutkin’s method of proof uses a famous identity involving topological entropy due to Mañé [28]: htop = lim T!1 logπM⇥M 1 T nT(x, y) dµ(x)dµ(y) (5.2.1) If a Riemannian manifold has no conjugate point this identity simply implies the following stronger identity: 8x, y 2 M, htop = lim T!1 1 T log nT(x, y) (5.2.2) which is then applied to obtain the aforementioned results. Therefore, a main approach for proving the main conjecture concerns studying a weaker version of Equation 5.2.2, and a modified technique to obtain similar results for manifolds with conjugate points. A big step in this regard is to understand behavior of the exponential map near singularities, which is mainly unknown. A famous paper by Warner [36], gives a very good insight about the conjugate locus and the type of conjugate points. Regarding mT(x, y) (or nT(x, y)) functions, I believe the following conjecture to be true which is an interesting problem by itself: 51 Conjecture 5.1. Let (M, g) be a closed connected Riemannian manifold. For T > 0, x, y 2 M, let mT(x, y) 1 be the cardinality of the set of geodesic segments connecting x to y of length  T. Then there exists a pair of points x⇤, y⇤ 2 M such that mT(x⇤, y⇤) = sup x,y2M mT(x, y) . Since M is compact and mT(x, y) is a discrete function, the statement of Conjecture 5.1 is trivial if supx,y2M mT(x, y) < 1. If supx,y2M mT(x, y) = 1, let (xn, yn) be a sequence in M ⇥ M such that mT(xn, yn)! 1 , as n ! 1. Since M ⇥ M is compact, passing to a subsequence if necessary, we may assume there exists (x⇤, y⇤)2 M ⇥ M such that (xn, yn)!( x⇤, y⇤), as n ! 1. One approach for the Conjecture is to assume mT(x⇤, y⇤) < 1 and that there exists a sequence yn ! y⇤, where mT(x⇤, yn)! 1 , as n ! 1 and derive a contradiction. Let expT be the exponential map at x⇤, restricted to a closed ball of radius T in the tangent space Tx⇤ M. Assume expT(p) = y⇤ and expT(pj n) = yn, j = 1, · · · , jn, where jn ! 1, as n ! 1. Interestingly, it can be proved (passing to a subsequence if necessary) that we can interpolate all pj n’s through a smooth path v : [0, a]! Tx M starting at p, so the parametrized surface f(s, t) = expT(tv(s)), 0  s  a, 0  t  1, would be a strange surface in M and its boundary c(s) = expT(v(s)), winds around itself a lot of times starting at y⇤. It would be interesting to study the Jacobi field Js(t) = @s f(s, t) of this surface, where a dierent geometric or analytic technique may be applied to possibly refute the existence of such surface. If proved, Conjecture 5.1 together with Mañé identity 5.2.1, has interesting implications re- garding mT(x, y) estimates which may also be advantageous for proving general Conjecture 1.1 for a certain class of manifolds with conjugate points. Conjecture 5.1, basically states that if expT is finite to one, that is the preimage of every point has finite cardinality) then it is a finite map at every point of its domain (See Golubitsky [10, pp. 167-169]). For a Riemnnian manifold M and a point x 2 M, a conjugate point p 2 Tx M is called regular if there exists a neighborhood U of p such that each ray of Tx M contains at most one point in U which is a conjugate point. A conjugate point which is not regular is called a singular or intersection point [36]. Warner [36, Theorem 52 3.3] proves for almost every regular point, the map expT can be formulated in local coordinates, which would imply the finiteness of expT at these points. However, for a class of regular conjugate points, and subsequently singular conjugate points proving the finiteness would be tough. There is also much room to investigate behavior of the exponential map near such points using singularity theory, and derive properties beyond just the finiteness. 53 APPENDICES 54 APPENDIX A Semidirect Product and Semidirect Sum In many cases it is convenient to describe the structure of Lie groups in terms of semidirect products. The semidirect product of abstract groups G1 and G2 is the direct product of sets G1 and G2 endowed with the group structure via (g1, g2)(h1, h2) = (g1 · b(g2)h1, g2h2) , where b is a homomorphism of G2 into the group AutG1 of automorphisms of the group G1. We will denote the semidirect product by G1 o G2, or more precisely G1 ob G2. The elements of the form (g1, e) (resp. (e, g2)) form a subgroup in G1 oG2 isomorphic to G1 (resp. G2). This subgroup is usually identified with G1 (resp. G2). The subgroup G1 is normal and g2g1g1 2 = b(g2)g1, g1 2 G1, g2 2 G2 . (A.0.1) The subgroup G2 is normal if and only if b is trivial, i.e. b(G2) = e; in this case the semidirect product coincides with the direct product G1 ⇥ G2 . One says that a group G splits into a semidirect product of subgroups G1 and G2 if 1. G1 is normal; 2. G1G2 = G; 3. G1 \ G2 = {e}. In this case we have the isomorphism G1 ob G2 G, (g1, g2) 7! g1g2 , (A.0.2) where b : G2 ! Aut G1 is the homomorphism defined by A.0.1 and we will write G = G1 o G2. 55 A semidirect product of Lie groups G1 and G2 is defined as a semidirect product of abstract groups endowed with a dierentiable structure as the direct product of dierentiable manifolds. It is additionally required that b define dierentiable G2-action on G1 , i.e. that the map G1 ⇥ G2 ! G1, (g1, g2) 7! b(g2)g1 be dierentiable. (In particular, the automorphism b(g2) of G1 must be dierentiable for any g2 2 G2). This ensures the dierentiability of group actions in the semidirect product. One says that a Lie group G splits into a semidirect product of Lie subgroups G1 and G2 if it splits into their semidirect product as an abstract group. In this case the action b of G2 on G1 defined by A.0.1 is dierentiable and the abstract isomorphism A.0.2 is a Lie group isomorphism. To semidirect products of Lie groups there correspond semidirect sums of Lie algebras (which could as well have been called semidirect products). The tangent algebra of Aut g is the Lie algebra der g of derivations of g, i.e. linear transformation of g satisfying the product rule (see Onishchick [32, p.23]). Let be a Lie algebra homomorphism g2 ! der g1. A semidirect sum of Lie algebras g1 and g2 is the direct sum of vector spaces g1 and g2 endowed with the bracket [(⇠1,⇠ 2),(⌘1,⌘ 2)] = ([⇠1,⌘ 1] + (⇠2)⌘1 (⌘2)⇠1,[⇠2,⌘ 2]) . We denote the semidirect sum by g1 + g2, or more prudently g1 + g2. It is not dicult to verify a semidirect sum of Lie algebras is a Lie algebra. The elements of the form (⇠1, 0) (resp. (0,⇠ 2)) constitute a subalgebra of g1 + g2 isomorphic to g1 (resp. g2), usually identified with g1 (resp. g2). The subalgebra g1 is an ideal and [⇠2,⇠ 1] = (⇠2)⇠1, (⇠1 2 g1,⇠ 2 2 g2) . (A.0.3) The subalgebra g2 is an ideal if and only if = 0. In this case the semidirect sum is isomorphic to the direct sum g1 g2. One says that a Lie algebra g splits into a semidirect sum of Lie subalgebras g1 and g2 if 1. g1 is an ideal; 56 2. g is the direct sum of subspaces g1 and g2 as a vector space. In this case we have an isomorphism g1 + g2 g, (⇠1,⇠ 2) 7! ⇠1 + ⇠2 , where : g2 ! der g1 is the homomorphism defined by formula A.0.3. In this situation we will write g = g1 + g2. The following theorem relates semidirect product of two Lie groups to semidirect sum of their corresponding Lie algebra, [32, p.37]. Theorem A.1. The tangent Lie algebra of the semidirect product G1 ob G2 of Lie groups G1 and G2 is the semidirect sum g1 + g2 of their tangent algebras and = dB, where B : G2 ! Aut(g1) is a Lie group homomorphism defined by the formula B(g2) = d(b(g2)), for any g2 2 G2· 57 APPENDIX B Levi Decomposition Let G be a connected Lie group and g = TeG its Lie algebra. Recall that the iterated commutator groups G(k) (k = 0, 1, 2, · · ·) of G are defined by induction: G0 = G, G(1) = G0, G(k+1) =⇣G(k)⌘0 . A Lie group G is called solvabale if there exists an integer m such that G(m) = {e}. The derived algebra of a Lie algebra g is the subalgebra [g, g] = g0 generated by the brackets [⇠, ⌘], where ⇠, ⌘ 2 g. It is the smallest ideal such that the corresponding quotient algebra is commutative. The iterated derived algebras g(k) (k = 0, 1, 2, · · ·) of a Lie aglebra g are defined by induction as: g0 = g, g(1) = g0, g(k+1) =⇣g(k)⌘0 . A Lie algebra is called solvable if there exists an m such that g(m) = {0}. A connected Lie group G is solvable if and only if so is its Lie algebra. More precisely, G(m) = {e} if and only if g(m) = {0}, Onishchick [32, p.54]. The sum of solvable ideals of a Lie algebra is a solvable ideal [32, p.55]. It follows that in any Lie algebra g there exists the largest solvable ideal. It is called the radical of g. We denote it by rad g. Similarly, the largest connected solvable normal Lie subgroup of G is called radical of the Lie group G and is denoted by Rad G. The following theorem guarantees existence of such subgroup (see [32, p.55] for proof). Theorem B.1. In any Lie group G there is the largest connected solvable normal Lie subgroup. Its tangent Lie algebra coincides with rad g. Examples. 58 1. The group Bn(K) the set of invertible (upper) triangular n ⇥ n matrices over the field K is a solvable Lie group, and its Lie algebra bn(K), the set of all upper triangular n ⇥ n matrices over K is a solvable Lie algebra [32, p.53]. 2. It is well known that semidirect product of two solvable abstract group is solvable. Thus Sol is solvable, so is its Lie algebra sol. 3. Nilpotent Lie groups (resp. Lie algebras) are, a fortiori solvable but the converse is not true, Hall [19, p.54]. A Lie group G (resp. a Lie algebra g) is called semisimple if Rad G = {e} (resp. rad g = {0}). Obviously, a Lie group is semisimple if and only if its tangent Lie algebra is semisimple. For any Lie group G (resp. Lie algebra g) the quotient group G/RadG (resp. the quotient algebra g/rad g) is semisimple. Equivalently, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., there exist non-abelian Lie algebras gi, i = 1, · · · , n whose only ideals of gi are 0 and gi itself, and g = g1 · · · gn [19, p.173]. Remark. A connected non-Abelian Lie group G is simple if it has no nontrivial, connected, closed, proper, normal subgroup. A non-abelian Lie algebra g is simple if its only ideals are 0 and itself (or equivalently, a Lie algebra of dimension 2 or more, whose only ideals are 0 and itself). It can be shown that G = SL(n, R), n > 1 (resp. so(n, R), n > 1) is a simple Lie group (resp. Lie algebra). A direct product of simple Lie groups (ex. SL(2, R)⇥ SL(3, R)) is semisimple. But in general, semisimple Lie groups are a much larger class of Lie groups. see Morris [31, p.428] for a definition based on direct product of simple Lie groups. The solvable Lie algebras and the semisimple Lie algebras form two large and generally com- plementary classes, as is shown by Levi decomposition. Definition B.2. Let g be a finite-dimensional Lie algebra over K = C or R. A subalgebra l ⇢ g is called a Levi subalgebra if g splits into the semidirect sum g = rad g + l . 59 (B.0.1) Decomposition B.0.1 is called the Levi decomposition of g. Theorem B.3 (Levi). Any finite dimensional Lie algebra g over K = C or R contains a Levi subalgebra. Analogous statements hold for simply connected Lie groups. The following theorem states one of the fundamental facts of Lie theory which sometimes is called Lie’s third theorem (See Onishchick [32, p.284] for proof). Theorem B.4. Let g be a finite-dimensional Lie algebra (over C or R), l its Levi subalgebra. Then there exists a simply connected Lie group G (either complex or real respectively) whose tangent algebra is isomorphic to g. Moreover, G = A o L , where A = Rad G, and L is a simply connected Lie subgroup with the tangent Lie algebra l. Remark. Rad G (resp. rad g) is solvable. In addition, for any Lie group G (resp. Lie algebra g) the quotient group G/RadG (resp. the quotient algebra g/rad g) is semisimple. Therefore Levi decompostion implies every simply connected Lie group (resp. finite-dimensional Lie algebra) is a semidirect product (resp. semidirect sum) of a solvable Lie subgroup (resp. Lie subalgebra) and a semisimple Lie subgroup (resp. Lie subaglebra). 60 APPENDIX C Haar Measure Standard texts in real analysis construct a translation-invariant measure on Rn which is called Lebesgue measure, but the analogue for the Lie groups is called Haar measure. We state the following proposition without proof. Proposition C.1 (Existence and uniqueness of Haar measure). If G is any Lie group, then there exists a unique (up to scalar multiple) -finite borel measure µ on G, such that 1. µ(C) is finite for every compact subset C of G, and 2. µ(gA) = µ(A), for every Borel subset A of G, and every g 2 G. Definition C.2. The measure µ of Proposition C.1 is called the left Haar measure on H. Analo- gously, there exists a unique right Haar measure with µ(Ag) = µ(A). G is unimodular if the left Haar measure is also a right Haar measure, i.e. µ(Ag) = µ(gA) = µ(A). Haar measure is always inner regular. This means µ(A) is the supremum of the measures of the compact subsets of A. Proposition C.3. There is a continuous homomorphism : G ! R+, such that, if µ is any (left or right) Haar measure on G, then µ(gAg1) = (h)µ(A), for all g 2 G and any Borel set A ⇢ G. Proof. Let µ be a left Haar measure. For each g 2 G, define h : G ! G by g(x) = gxg1. Then g is an automorphism of G, so (g)⇤ µ is a left Haar measure. By uniqueness, we conclude that there exists (g)2 R+, such that (g)⇤ µ = (g)µ. It is easy to see that is a continuous homomorphism. If µ is a left Haar measure it is easy to see that ˜µ(A) := µ(A1) is a right Haar measure and to verify the same formula also applies to it. ⇤ Definition C.4. The function defined in Proposition C.3 is called the modular function of G. 61 Corollary C.5. Let be the modular function of G, and let A be a Borel subset of G. i) If µ is a right Haar measure on G, then µ(gA) = (g)µ(A), for all g 2 G. ii) If µ is a left Haar measure on G, then µ(Ag) = (g1)µ(A), for all g 2 G. iii) G is unimodular if and only if (g) = 1, for all g 2 G. iv) (g) = | det(Adg)|, for all g 2 G. Proposition C.6. Let µ be a left Haar measure on a Lie group G. Then µ(G) < 1 if and only if G is compact. Proof. ((): See Proposition C.1 ()): Since µ(G) < 1 (and the measure µ is inner regular), there exists a compact subset C of G, such that µ(C) > µ(G)/2. Then, for any g 2 G, we have µ(gC) + µ(C) = µ(C) + µ(C) = 2µ(C) > µ(G) , so gC can not be disjoint from C. This implies that g belongs to the set C · C1, which is compact. Since g is an arbitrary element of G, we conclude that G = C · C1 is compact. ⇤ 62 APPENDIX D Arithmetic Subgroups A lattice of the form GZ = G \ SL(n, Z) is said to be arithmetic. However, for the following reasons, a somewhat more general class of lattices is also said to be arithmetic. The reason is that there are some obvious modifications of GZ that are also lattices, and they should also be regarded as arithmetic subgroups. We want to modify the definition so that: 1. If : G1 ! G2 is an isomorphism, and 1 is an arithmetic subgroup of G1, then we wish to be able to say that (1) is an arithmetic subgroup of G2. 2. We wish to ignore compact groups, that is, modding out a compact subgroup should not aect arithmeticity. So we wish to be able that if K is a compact normal subgroup of G, and is a lattice in G, then is arithmetic if and only if K/K is an arithmetic subgroup of G/K. 3. And finally, arithmeticity should be independent of commensurability. First we need the following definition. Definition D.1. For a subset Q of Q[x1,1, · · · , xn,n], we define Var(Q) := {g 2 SL(n, R) | Q(g) = 0, for all Q 2Q} , which is a subgroup of SL(n, R). Let H be a closed subgroup of SL(n, R). We say that H is defined over Q (or that H is a Q-subgroup) if there exists a subset Q⇢ Q[x1,1, · · · , xn,n] such that H = Var(Q), and H has only finitely many components. In other words, H is commensurable to the variety Var(Q), for some set Q of Q-polynomials. Examples. 1. SL(n, R) is defined over Q, just let Var(Q) = ;. 63 2. If m < n, we may embed SL(m, R) in the top left corner of SL(n, R). This copy of SL(m, R) is defined over Q: Let Q = {xi,j j We state the following important theorem from Morris [31, p.88] without proof. i | max{i, j} > n}. Theorem D.2. If G is defined over Q, then GZ is lattice in G. This theorem immediately implies that SL(n, Z) is a lattice in SL(n, R). These considerations lead us to the following definition: Definition D.3. is an arithmetic subgroup of G if and only if there exist i) a closed, connected, semisimple subgroup G0 of some SL(n, R), such that G0 is defined over Q, ii) compact normal subgroups K and K0 of G and G0, respectively, and iii) an isomorphism : G/K ! G0/K0, such that () is commensurable to G0Z, where and G0Z are the images of \ G and G0Z in G/K and G0/K0, respectively. SL(n, Z) is the most basic example of an arithmetic group. In Section 4.3 we present a quater- nionic structure of SL(2, R) and we make arithmetic subgroups SL(1, Ha,b Z ) which are cocompact lattices of SL(2, R). Note that up to conjugacy, there are only countably many arithmetic lattices in G, because there are only countably many finite subsets of the polynomial ring Q[x1,1, · · · , xn,n]. The following theorem is a very helpful criterion for arithmeticity. Recall that the subgroup CommG() = {g 2 G | gg1 is commensuarble to } is called the commensurator of in G. GQ ⇢ CommG(GZ). It is easy to see that if G is defined over Q, then For a connected semisimple non-compact Lie group G, a lattice ⇢ G is called irreducible if for every non-compact closed normal subgroup N of G, N is dense in G. In particular, lattices of 64 the form 1 ⇥ 2 ⇢ G1 ⇥ G2 are excluded from this definition and are called reducible. Note that SL(n, R) is a simple Lie group, thus all lattices ⇢ SL(n, R) are irreducible. Theorem D.4 (Commensurability Criterion for Arithmeticity). Let G be a connected semisimple Lie group with no compact factors, and ⇢ G an irreducible lattice. Then is arithmetic if and only if CommG() of is dense in G. The astonishing theorem due to Gregory Margulis shows that taking the integer points is usually the only way to make a lattice. If G is a semisimple algebraic group defined over the field F, then F-rank(G) is defined to be the maximal dimension of an abelian F-subgroup of G which is F- split, i.e. which can be diagonalized over F. If G is a connected semisimple Lie group then we can realize Ad(G) as a subgroup of finite index in the R-points of an R-group (See Zimmer [38, Proposition 3.1.6]). We then define R-rank(G) to be the R-rank of this algebraic group. Thus R-rank(SL(n, R)) = n 1, the R-split abelian subgroup of maximal dimension being diagonal matrices of dimension one. Theorem D.5 (Margulis Arithmeticity). Let G be a connected semisimple Lie group with trivial center and no compact factors. Let ⇢ G be an irreducible lattice. Assume R-rank(G) 2. Then is arithmetic. 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