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We observe that the strong slope conjecture implies that the degree of the colored Jones polynomial detects all torus knots. As an application we obtain that an adequate knot that has the same colored Jones polynomial degrees as a torus knot must be a $(2,q)$torus knot.

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.

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 obtain a formula for the TuraevViro invariants of a link complement in terms of values of the colored Jones polynomial of the link. As an application we give the first examples for which the volume conjecture of Chen and the third named author\,\cite{ChenYang} is verified. Namely, we show that the asymptotics of the TuraevViro invariants of the Figureeight knot and the Borromean rings complement determine the corresponding hyperbolic volumes. Our calculations also exhibit new phenomena of asymptotic behavior of values of the colored Jones polynomials that seem not to be predicted by neither the KashaevMurakamiMurakami volume conjecture and various of its generalizations nor by Zagier's quantum modularity conjecture. We conjecture that the asymptotics of the TuraevViro invariants of any link complement determine the simplicial volume of the link, and verify it for all knots with zero simplicial volume. Finally we observe that our simplicial volume conjecture is stable under connect sum and split unions of links.