Non-realizability of some big mapping class groups

TL;DR

This paper demonstrates that certain big mapping class groups, including those of surfaces with genus 3 subsurfaces or specific symmetries, cannot be realized as subgroups of the homeomorphism group.

Contribution

It establishes non-realizability results for large classes of mapping class groups, extending understanding of their algebraic and geometric properties.

Findings

Mapping class group of genus 3 subsurface not realizable as homeomorphism subgroup

Certain symmetric surfaces' mapping class groups are not realizable as homeomorphism subgroups

Examples include the plane minus a Cantor set and the sphere minus a Cantor set

Abstract

In this note, we prove that the compactly supported mapping class group of a surface containing a genus $3$ subsurface has no realization as a subgroup of the homeomorphism group. We also prove that for certain surfaces with order $6$ symmetries, their mapping class groups have no realization as a subgroup of the homeomorphism group. Examples of such surfaces include the plane minus a Cantor set and the sphere minus a Cantor set.