Three perfect mapping class groups

Abstract

We prove that the mapping class group of a surface obtained from removing a Cantor set from either the 2-sphere, the plane, or the interior of the closed 2-disk has no proper countable-index subgroups. The proof is an application of the automatic continuity of these groups, which was established by Mann. As corollaries, we see that these groups do not contain any proper finite-index subgroups and that each of these groups have trivial abelianization.

v2: updated/corrected references, corrected abstract (which was missing a key adjective), 6 pages

References in corpus (3)

• Big mapping class groups: an overview
• Large scale geometry of big mapping class groups
• Automatic continuity for homeomorphism groups of noncompact manifolds

Cited by in corpus (1)

• Self-similar surfaces: involutions and perfection