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Growth of quantum $6j$-symbols and applications to the Volume Conjecture.
We prove the Turaev-Viro invariants volume conjecture for a "universal" class of cusped hyperbolic 3-manifolds that produces all 3-manifolds with empty or toroidal boundary by Dehn filling. This leads to two-sided bounds on the volume of any hyperbolic 3-manifold with empty or toroidal boundary in terms of the growth rate of the Turaev-Viro invariants of the complement of an appropriate link contained in the manifold. We also provide evidence for a conjecture of Andersen, Masbaum and Ueno (AMU conjecture) about certain quantum representations of surface mapping class groups. A key step in our proofs is finding a sharp upper bound on the growth rate of the quantum 6j−symbol evaluated at q=e2πir.
Authors:
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Award ID(s):
Publication Date:
NSF-PAR ID:
10323851
Journal Name:
Journal of differential geometry
Volume:
120
Issue:
2
Page Range or eLocation-ID:
199-229
ISSN:
0022-040X
2. We establish a relation between the "large r" asymptotics of the Turaev-Viro invariants $TV_r$and the Gromov norm of 3-manifolds. We show that for any orientable, compact 3-manifold $M$, with (possibly empty) toroidal boundary, $log|TVr(M)|$ is bounded above by a function linear in $r$ and whose slope is a positive universal constant times the Gromov norm of $M$. The proof combines TQFT techniques, geometric decomposition theory of 3-manifolds and analytical estimates of $6j$-symbols. We obtain topological criteria that can be used to check whether the growth is actually exponential; that is one has $log|TVr(M)|\geq B r$, for some $B>0$. Wemore »