Power quotients of surface groups and mapping class groups

TL;DR

This paper investigates the structure of power quotients of surface groups, establishing an isomorphism with mapping class groups modulo powers of Dehn twists and exploring their algebraic and geometric properties.

Contribution

It provides an analogue of the Dehn--Nielsen--Baer theorem for large n, describing automorphism groups of surface group quotients in terms of mapping class groups.

Findings

Outer automorphism group of $ ext{Gamma}(n)$ is isomorphic to the quotient of the extended mapping class group by n-th powers of Dehn twists.

$ ext{Gamma}(n)$ is virtually torsion-free, acylindrically hyperbolic, and has solvable word problem.

The groups have finite asymptotic dimension and are infinitely presented.

Abstract

Let $\Gamma$ be the fundamental group of a closed, orientable, hyperbolic surface $S$. The $n$-power quotient, $\Gamma(n)$, is the quotient of $\Gamma$ by the $n$th powers of simple closed curves. We prove an analogue of the Dehn--Nielsen--Baer theorem for suitable large values of $n$: the outer automorphism group of $\Gamma(n)$ is isomorphic to the quotient of the extended mapping class group of $S$ by $n$th powers of Dehn twists. There is also a corresponding description of the automorphism group as the quotient of the extended mapping class group of the corresponding once-punctured surface, and we relate these groups via a Birman-type exact sequence. Along the way, and as consequences, we prove structural properties of $\Gamma(n)$ for suitable large values of $n$, including: $\Gamma(n)$ is virtually torsion-free, acylindrically hyperbolic, infinitely presented, with solvable word problem and finite asymptotic dimension.