The Grothendieck-Teichm\"uller group $\widehat{GT}$ acts on the genus $g$ mapping class group with $0$ or $1$ marked point

TL;DR

This paper proves that the Grothendieck-Teichmüller group acts on the profinite genus g mapping class groups with zero or one marked point for all g > 0, revealing new symmetries in algebraic topology.

Contribution

It establishes the action of the Grothendieck-Teichmüller group on profinite mapping class groups with marked points, extending understanding of their symmetries.

Findings

Grothendieck-Teichmüller group acts on profinite genus g mapping class groups

The action is proven for all g > 0

Provides new insights into the structure of mapping class groups

Abstract

The goal of this article is to prove that the Grothendieck-Teichm\"uller group $\widehat{GT}$ acts on $\widehat{\Gamma}_{g,0}$ and $\widehat{\Gamma}_{g,1}$,the (full) profinite genus $g$ mapping class group with $0$ or $1$ marked point, for every $g>0$.