On abstract commensurators of surface groups

TL;DR

This paper investigates the structure of the abstract commensurator of surface groups, revealing the presence of Baumslag--Solitar groups and their properties, using computational methods to analyze finitely-generated subgroups.

Contribution

It demonstrates that the abstract commensurator contains solvable Baumslag--Solitar groups and provides computational proofs of their properties, such as non-residual finiteness.

Findings

Comm$(mma)$ contains Baumslag--Solitar groups for any n > 1

The group ngle a, b : a b^2 a^{-1} = b^3 angle is not residually finite

Finitely-generated subgroups of Comm$(mma)$ are amenable to computational analysis

Abstract

Let $\Gamma$ be the fundamental group of a surface of finite type and Comm$(\Gamma)$ be its abstract commensurator. Then Comm$(\Gamma)$ contains the solvable Baumslag--Solitar groups $\langle a ,b : a b a^{-1} = b^n \rangle$ for any $n > 1$. Moreover, the Baumslag--Solitar group $\langle a ,b : a b^2 a^{-1} = b^3 \rangle$ has an image in Comm$(\Gamma)$ that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of Comm$(\Gamma)$ are concrete objects amenable to computational methods. For example, we give a proof that $\langle a ,b : a b^2 a^{-1} = b^3 \rangle$ is not residually finite without the use of normal forms of HNN extensions.