Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudoAnosov mapping classes to elements of infinite order (for large enough level r). In this paper, we relate the AMU conjecture to a question about the growth of the TuraevViro invariants TVr of hyperbolic 3manifolds. We show that if the rgrowth of TVr(M) for a hyperbolic 3manifold M that fibers over the circle is exponential, then the monodromy of the fibration of M satisfies the AMU conjecture. Building on earlier work \cite{DK} we give broad constructions of (oriented) hyperbolic fibered links, of arbitrarily high genus, whose SO(3)TuraevViro invariants have exponential rgrowth. As a result, for any g>n⩾2, we obtain infinite families of nonconjugate pseudoAnosov mapping classes, acting on surfaces of genus g and n boundary components, that satisfy the AMU conjecture. We also discuss integrality properties of the traces of quantum representations and we answer a question of Chen and Yang about TuraevViro invariants of torus links.
Gromov norm and TuraevViro invariants of 3manifolds
We establish a relation between the "large r" asymptotics of the TuraevViro invariants $TV_r $and the Gromov norm of 3manifolds. We show that for any orientable, compact 3manifold $M$, with (possibly empty) toroidal boundary, $logTVr(M)$ 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 3manifolds 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 $logTVr(M)\geq B r$, for some $B>0$. We use these criteria to construct infinite families of hyperbolic 3manifolds whose $SO(3)$ TuraevViro invariants grow exponentially. These constructions are essential for the results of article [3] where we make progress on a conjecture of Andersen, Masbaum and Ueno about the geometric properties of surface mapping class groups detected by the quantum representations.
We also study the behavior of the TuraevViro invariants under cutting and gluing of 3manifolds along tori. In particular, we show that, like the Gromov norm, the values of the invariants do not increase under Dehn filling and we give applications of this result on the question of more »
 Award ID(s):
 2004155
 Publication Date:
 NSFPAR ID:
 10233267
 Journal Name:
 Annales scientifiques de lÉcole normale supérieure
 Volume:
 53
 Issue:
 6
 Page Range or eLocationID:
 1363–1391
 ISSN:
 00129593
 Sponsoring Org:
 National Science Foundation
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Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudoAnosov mapping classes to elements of infinite order (for large enough level r). In this paper, we relate the AMU conjecture to a question about the growth of the TuraevViro invariants TVr of hyperbolic 3manifolds. We show that if the rgrowth of TVr(M) for a hyperbolic 3manifold M that fibers over the circle is exponential, then the monodromy of the fibration of M satisfies the AMU conjecture. Building on earlier work \cite{DK} we give broad constructions of (oriented) hyperbolic fibered links, of arbitrarily high genus, whose SO(3)TuraevViro invariants have exponential rgrowth. As a result, for any g>n⩾2, we obtain infinite families of nonconjugate pseudoAnosov mapping classes, acting on surfaces of genus g and n boundary components, that satisfy the AMU conjecture. We also discuss integrality properties of the traces of quantum representations and we answer a question of Chen and Yang about TuraevViro invariants of torus links.

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.

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.

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