Quotients of the mapping class group by power subgroups

TL;DR

This paper investigates the structure of quotients of the mapping class group by subgroups generated by p-th powers of Dehn twists, revealing complex properties such as the presence of infinite normal subgroups and Kähler subgroups, depending on the surface and p.

Contribution

It provides new insights into the algebraic and geometric properties of these quotients, including conditions for the existence of Kähler subgroups and the structure of their normal subgroups.

Findings

The quotient contains an infinite normal subgroup of infinite index for most p.

The quotient has a Kähler subgroup of finite index when p is coprime with six.

Finite-index subgroups with infinite abelianization relate to the quotient's structure.

Abstract

We study the quotient of the mapping class group $\operatorname{Mod}_g^n$ of a surface of genus $g$ with $n$ punctures, by the subgroup $\operatorname{Mod}_g^n[p]$ generated by the $p$-th powers of Dehn twists. Our first main result is that $\operatorname{Mod}_g^1 /\operatorname{Mod}_g^1[p]$ contains an infinite normal subgroup of infinite index, and in particular is not commensurable to a higher-rank lattice, for all but finitely many explicit values of $p$. Next, we prove that $\operatorname{Mod}_g^0/ \operatorname{Mod}_g^0[p]$ contains a K\"ahler subgroup of finite index, for every $p\ge 2$ coprime with six. Finally, we observe that the existence of finite-index subgroups of $\operatorname{Mod}_g^0$ with infinite abelianization is equivalent to the analogous problem for $\operatorname{Mod}_g^0/ \operatorname{Mod}_g^0[p]$.