Mapping class groupoids and Thompson's groups

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 $\mathsf T$\mathrm{.}Wealsointroducelargergroupoid$$\mathbf{\Omega }\mathbf{M}\mathbf{G}$$, which is related to outer automorphisms of free groups and Thompson’s group $\mathsf{V}$ in a similar manner.