Non-planar ends are continuously unforgettable

TL;DR

The paper proves that certain subgroups of mapping class groups of infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms, leading to their classification as Hopfian topological groups.

Contribution

It establishes that continuous epimorphisms between these subgroups are always geometric, extending understanding of their structure and automorphisms.

Findings

Continuous epimorphisms are induced by homeomorphisms.

These groups are Hopfian topological groups.

Applicable to various subgroups including pure and full mapping class groups.

Abstract

We show that continuous epimorphisms between a class of subgroups of mapping class groups of orientable infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms. This class of subgroups includes the pure mapping class group, the closure of the compactly supported mapping classes, and the full mapping class group in the case that the underlying manifold has a finite number of ends or is perfectly self-similar. As a corollary, these groups are Hopfian topological groups.