Homeomorphism groups of telescoping $2$-manifolds are strongly distorted

Abstract

Building on the work of Mann and Rafi, we introduce an expanded definition of a telescoping $2$-manifold and proceed to study the homeomorphism group of a telescoping $2$-manifold. Our main result shows that it is strongly distorted. We then give a simple description of its commutator subgroup, which is index one, two, or four depending on the topology of the manifold. Moreover, we show its commutator subgroup is uniformly perfect with commutator width at most two, and we give a family of uniform normal generators for its commutator subgroup. As a consequence of the latter result, we show that for an (orientable) telescoping $2$-manifold, every (orientation-preserving) homeomorphism can be expressed as a product of at most 17 conjugate involutions. Finally, we provide analogous statements for mapping class groups.