On abstract commensurators of surface groups

Let $\mathrm{\Gamma }$ be the fundamental group of a surface of finite type and Comm$(\mathrm{\Gamma })$ be its abstract commensurator. Then Comm$(\mathrm{\Gamma })$ contains the solvable Baumslag–Solitar groups $⟨a,b:ab{a}^{-1}=b^n⟩$ for any $n>1$. Moreover, the Baumslag–Solitar group $\langle a ,b : a b^2 a^{-1} = b^3 \rangle$hasanimageinComm$(\Gamma)$ that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of Comm$(\mathrm{\Gamma })$ 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.