Abstract commensurators of surface groups

Let be the fundamental group of a surface of finite type and Comm be its abstract commensurator. Then Comm contains the solvable Baumslag-Solitar groups (a,b: Aba-1 = bn) for any n > 1. Moreover, the Baumslag-Solitar group (a,b: Ab2a-1 = b3) has an image in Comm that is not residually finite. Our proofs are computer-Assisted. Our results also illustrate that finitely-generated subgroups of Comm are concrete objects amenable to computational methods. For example, we give a proof that (a,b: Ab2a-1 = b3) is not residually finite without the use of normal forms of HNN extensions.