Cosets of monodromies and quantum representations

Abstract

We use geometric methods to show that given any 3-manifold M , and g a sufficiently large integer, the mapping class group Mod(Σ g,1 ) contains a coset of an abelian subgroup of rank g 2 , consisting of pseudo-Anosov monodromies of open-book decompositions in M. We prove a similar result for rank two free cosets of Mod(Σ g,1 ). These results have applications to a conjecture of Andersen, Masbaum and Ueno about quantum representations of surface mapping class groups. For surfaces with boundary, and large enough genus, we construct cosets of abelian and free subgroups of their mapping class groups consisting of elements that satisfy the conjecture. The mapping tori of these elements are fibered 3-manifolds that satisfy a weak form of the Turaev-Viro invariants volume conjecture.

Fundamental shadow links realized as links in $S^3$

“…In [20], Marché and Santharoubane construct finitely many conjugacy classes of pseudo-Anosov elements for M od(Σ g,1 ), and Detcherry and Kalfagianni construct cosets of M od(Σ g,1 ) for sufficiently large g that satisfy Conjecture 1.3 in [10]. Now we remark that two elements f, g ∈ M od(Σ g,n ) are independent if there is no h ∈ M od(Σ g,n ) such that both f and g are not conjugate to non-trivial powers of h. As explained in [10], the motivation behind the definition of independent is if f and g are not independent, then f satisfies the AMU conjecture if and only if g does. It has been shown by Detcherry and Kalfagianni in [12] that there are infinitely many pairwise independent pseudo-Anosov elements in M od(Σ g,2 ) for g ≥ 3 and for M od(Σ g,n ) where g ≥ n ≥ 3.…”