Power quotients of surface groups and mapping class groups

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.