We study asymptotic properties of the Turaev--Viro invariants and build on their conjecturedconnections to the geometry of 3-manifolds. Our focus is a conjecture of Chen and Yang asserting that the exponential growth rate of the Turaev--Viro invariants coincides with the hyperbolic volume of the manifold. We begin by studying the variation of theinvariants of a manifold with toroidal boundary under the operation of attaching a (p,q)-torus knot cable space, establishing that the Chen--Yang conjecture is stable under this operation and generalizing results of Detcherry. In doing so, we make heavy use of the SO(3)-Reshetikhin--Turaev TQFTs and their relationship to the Turaev--Viro invariants.Next we introduce an infinite family of link complements, constructed from gluings of elementaryhyperbolic manifolds inspired by work of Agol, satisfying the Chen--Yang volume conjecture. We show the asymptotics of the Turaev--Viro invariants of these manifolds are additive under the gluings of the elementary pieces, giving the first examples satisfying the conjecture which have an arbitrary number of hyperbolic pieces. Lastly, we study a weak form of the Chen--Yang conjecture known as the Exponential GrowthConjecture, which asserts that the exponential growth rate of the Turaev--Viro invariants is positive. We construct the first infinite families of knots in S^3 satisfying this conjecture using Dehn surgery methods. Detcherry and Kalfagianni show the Exponential Growth Conjecture implies a conjecture of Andersen, Masbaum, and Ueno. Using this, we construct an infinite family of mapping classes acting on surfaces of any genus and one boundary component satisfying the conjecture of Andersen, Masbaum, and Ueno which correspond to fibered knots in S^3.
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