On abstract commensurators of surface groups

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 $⟨a,b:ab{a}^{-1}=b^n⟩$ for any $n>1$. Moreover, the Baumslag--Solitar group $⟨a,b:a{b}^2{a}^{-1}=b^3⟩$ 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 $⟨a,b:a{b}^2{a}^{-1}=b^3⟩$ is not residually finite without the use of normal forms of HNN extensions.

14 pages, 1 figure. v2: Substantial revision in response to comments from Yves Cornulier, with some added new results. v3: Restructured, following referee comments. Title changed. Accepted for publication in Journal of Topology and Analysis