Finite presentations of centrally extended mapping class groups

TL;DR

This paper provides a finite presentation for the universal central extension of the mapping class group of surfaces with genus at least 3, enhancing understanding of their algebraic structure.

Contribution

It offers the first explicit finite presentation of the universal central extension of mapping class groups for genus g ≥ 3 surfaces.

Findings

Finite presentation for $\mathcal{T}_{g,r}$ with $g \geq 3$

Analysis of the case $g=2$

Advances algebraic understanding of mapping class groups

Abstract

We describe a finite presentation of $ \mathcal{T}_{g,r} $ for $g \geq 3$. % or $(g,r)=(2,0)$. Here $\mathcal{T}_{g,r} $ is the universal central extension of the mapping class group of the surface of genus $g$ with $r$-boundaries. We also investigate the case $g=2$,