Mapping class groupoids and Thompson's groups

TL;DR

This paper introduces new groupoids related to mapping class groups and Thompson's groups, exploring their connections to surface automorphisms and automorphisms of free groups.

Contribution

It defines the fundamental modular groupoid and a larger groupoid, establishing their relationships with surface mapping class groups, outer automorphisms of free groups, and Thompson's groups.

Findings

Defined the fundamental modular groupoid and its properties.

Established connections between the groupoids and Thompson's groups.

Explored the relationship with surface mapping class groups and automorphisms.

Abstract

We introduce a groupoid ${\mathbf{\Pi MG}}}$, called the fundamental modular groupoid, which is a variant of Penner's mapping class groupoid. We study how it relates to the surface mapping class groups and Thompson's group $\mathsf T$. We also introduce larger groupoid $\mathbf{\Omega MG}$, which is related to outer automorphisms of free groups and Thompson's group $\mathsf V$ in a similar manner.