Quantum representations and monodromies of fibered links

TL;DR

This paper explores the connection between quantum representations of surface mapping class groups, hyperbolic 3-manifold invariants, and the AMU conjecture, providing new constructions of fibered links with exponential Turaev-Viro invariant growth.

Contribution

It relates the AMU conjecture to Turaev-Viro invariants growth and constructs hyperbolic fibered links with exponential invariant growth, supporting the conjecture for broad classes of mapping classes.

Findings

Exponential growth of Turaev-Viro invariants implies the AMU conjecture for certain fibered 3-manifolds.

Constructed infinite families of hyperbolic fibered links with high genus and exponential Turaev-Viro invariants.

Provided insights into the traces of quantum representations and answered questions about Turaev-Viro invariants of torus links.

Abstract

Andersen, 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 $TV_r$ of hyperbolic 3-manifolds. We show that if the $r$-growth of $|TV_r(M)|$ 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 $g>n\geqslant 2$, 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.