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.
Cosets of monodromies and quantum representations
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.
 Award ID(s):
 1708249
 Publication Date:
 NSFPAR ID:
 10156676
 Journal Name:
 Advances in mathematics
 Volume:
 351
 Page Range or eLocationID:
 775813
 ISSN:
 00018708
 Sponsoring Org:
 National Science Foundation
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