On endomorphisms of surface mapping class groups

TL;DR

This paper proves that any endomorphism of the mapping class group of an orientable surface onto a finite index subgroup is actually an automorphism, revealing a strong rigidity property of these groups.

Contribution

It establishes a new rigidity result showing all endomorphisms onto finite index subgroups are automorphisms, extending understanding of the structure of surface mapping class groups.

Findings

Endomorphisms onto finite index subgroups are automorphisms.

Mapping class groups exhibit strong rigidity properties.

The result applies to all orientable surfaces.

Abstract

We prove that every endomorphism of the mapping class group of an orientable surface onto a subgroup of finite index is in fact an automorphism.