%ARenaud Detcherry, Efstratia%BJournal Name: Advances in mathematics; Journal Volume: 351
%D2019%I
%JJournal Name: Advances in mathematics; Journal Volume: 351
%K
%MOSTI ID: 10156676
%PMedium: X; Size: 775-813
%TCosets of monodromies and quantum representations
%XAndersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov 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 Turaev-Viro invariants TVr of hyperbolic 3-manifolds. We show that if the r-growth of |TVr(M)| for a hyperbolic 3-manifold 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)-Turaev-Viro invariants have exponential r-growth. As a result, for any g>n⩾2, we obtain infinite families of non-conjugate pseudo-Anosov 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 Turaev-Viro invariants of torus links.
%0Journal Article