Arithmetic quotients of the mapping class group

TL;DR

This paper constructs a broad family of arithmetic quotients of the mapping class group using representations of finite groups, revealing new structures beyond the classical symplectic quotient.

Contribution

It introduces a method to produce many new arithmetic quotients of the mapping class group from finite group representations, expanding understanding of its arithmetic properties.

Findings

Mapping class group has many new arithmetic quotients beyond the classical case.

These quotients are defined over various cyclotomic fields.

Existence of quotients of types Sp, SO, and SU for arbitrarily large dimensions.

Abstract

To every $Q$-irreducible representation $r$ of a finite group $H$, there corresponds a simple factor $A$ of $Q[H]$ with an involution $\tau$. To this pair $(A,\tau)$, we associate an arithmetic group $\Omega$ consisting of all $(2g-2)\times (2g-2)$ matrices over a natural order of $A^{op}$ which preserve a natural skew-Hermitian sesquilinear form on $A^{2g-2}$. We show that if $H$ is generated by less than $g$ elements, then $\Omega$ is a virtual quotient of the mapping class group $Mod(\Sigma_g)$, i.e. a finite index subgroup of $\Omega$ is a quotient of a finite index subgroup of $\Mod(\Sigma_g)$. This shows that the mapping class group has a rich family of arithmetic quotients (and "Torelli subgroups") for which the classical quotient $Sp(2g, Z)$ is just a first case in a list, the case corresponding to the trivial group $H$ and the trivial representation. Other pairs of $H$ and $r$ give rise to many new arithmetic quotients of $Mod(\Sigma_g)$ which are defined over various (subfields of) cyclotomic fields and are of type $Sp(2m), SO(2m,2m),$ and $SU(m,m)$ for arbitrarily large $m$.