Zariski density and finite quotients of mapping class groups

TL;DR

This paper proves that quantum representations of certain mapping class groups are Zariski dense in higher rank algebraic groups and uses this to produce many finite quotients, expanding understanding of their algebraic and finite structures.

Contribution

It establishes Zariski density of quantum representations of mapping class groups and constructs numerous finite quotients beyond symplectic groups.

Findings

Quantum representations are Zariski dense in higher rank algebraic groups.

Mapping class groups have many finite quotients outside symplectic groups.

Surjections onto finite groups rom the algebraic groups over finite rings.

Abstract

Our main result is that the image of the quantum representation of a central extension of the mapping class group of the genus $g\geq 3$ closed orientable surface at a prime $p\geq 5$ is a Zariski dense discrete subgroup of some higher rank algebraic semi-simple Lie group $\mathbb G_p$ defined over $\Q$. As an application we find that, for any prime $p\geq 5$ a central extension of the genus $g$ mapping class group surjects onto the finite groups $\mathbb G_p(\Z/q\Z)$, for all but finitely many primes $q$. This method provides infinitely many finite quotients of a given mapping class group outside the realm of symplectic groups.