We prove the TuraevViro invariants volume conjecture for a "universal" class of cusped hyperbolic 3manifolds that produces all 3manifolds with empty or toroidal boundary by Dehn filling. This leads to twosided bounds on the volume of any hyperbolic 3manifold with empty or toroidal boundary in terms of the growth rate of the TuraevViro 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 boundmore »
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Growth of quantum $6j$symbols and applications to the Volume Conjecture.
We prove the TuraevViro invariants volume conjecture for a "universal" class of cusped hyperbolic 3manifolds that produces all 3manifolds with empty or toroidal boundary by Dehn filling. This leads to twosided bounds on the volume of any hyperbolic 3manifold with empty or toroidal boundary in terms of the growth rate of the TuraevViro 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.
 Award ID(s):
 2004155
 Publication Date:
 NSFPAR ID:
 10323851
 Journal Name:
 Journal of differential geometry
 Volume:
 120
 Issue:
 2
 Page Range or eLocationID:
 199229
 ISSN:
 0022040X
 Sponsoring Org:
 National Science Foundation
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