Commensurators of surface braid groups

TL;DR

This paper establishes a natural isomorphism between the abstract commensurator of surface braid groups and the extended mapping class group of a related surface, deepening understanding of their algebraic and geometric structures.

Contribution

It proves that for surfaces of genus at least two, the abstract commensurator of the surface braid group is isomorphic to the extended mapping class group.

Findings

The abstract commensurator of surface braid groups is isomorphic to the extended mapping class group.

This result holds for surfaces with genus g ≥ 2 and n ≥ 2 strands.

Provides a new perspective on the algebraic structure of surface braid groups.

Abstract

We prove that if g and n are integers at least two, then the abstract commensurator of the braid group with n strands on a closed orientable surface of genus g is naturally isomorphic to the extended mapping class group of a compact orientable surface of genus g with n boundary components.