Isomorphisms and commensurability of surface Houghton groups (Q6611936)

Let \(\Sigma_{r}\) be the connected, orientable surface with empty boundary and exactly \(r\) ends, all of which are non-planar. The surface \(\Sigma_{r}\) can be viewed as constructed from a compact surface of genus \(g\) with \(r\) boundary components by inductively gluing copies of a surface of genus \(h\) with two boundary components, and then taking the union of the surfaces obtained at each step. The surface Houghton group \(B(g,h,r)\) defined by the above data is the subgroup of the mapping class group \(\mathrm{Map}(\Sigma_{r})\) whose elements are eventually rigid (for a more precise definition, see Section 2 of the paper).\N\NThe family of surface Houghton groups constructed here is more general than that introduced by the first author et al. [Math. Ann. 389, No. 4, 4301--4318 (2024; Zbl 1548.20069)], which corresponds to the case \(g=0\) and \(h=1\).\N\NIn the paper under review, the authors describe the commensurability and isomorphism classification of surface Houghton groups and their pure subgroups.