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

TL;DR

This work proves that the Grothendieck-Teichmüller group $\widehat{GT}$ acts by automorphisms on the full profinite genus-$g$ mapping class groups $\widehat{\Gamma}_{g,1}$ and $\widehat{\Gamma}_{g,0}$ for all $g>0$, extending Drinfeld's genus-zero braid action to higher genus. The authors provide an explicit automorphism on the generators $(a_1,\dots,a_{2g},d)$ via $F(a_1)=a_1^\lambda$, $F(d)=d^\lambda$, and $F(a_i)=f(a_i^2,y_i)\,a_i^\lambda\,f(y_i,a_i^2)$ with $y_i=a_{i-1}\cdots a_1^2\cdots a_{i-1}$, and verify this action preserves the Wajnryb relations (A), (B), (C), (C′), and, after introducing (D), descends to $\widehat{\Gamma}_{g,0}$. The proof proceeds by handling the genus-2 case, then genus-3 via lantern- and lantern-type arguments, and finally the higher-genus cases, with careful use of braid-group embeddings and the lantern relation to control the $f$-factors. These results connect the GT framework to the profinite mapping class groups, revealing a consistent automorphism tower consistent with Grothendieck’s lego philosophy and offering a pathway toward actions on $\widehat{\Gamma}_{g,n}$ for $n>1$ and related Lego-structured subgroups. Overall, the paper establishes a foundational GT-action on a broad class of profinite mapping class groups, with explicit formulas and structural verifications that ground future generalizations and applications in arithmetic topology.

Abstract

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

Figures (7)

• Figure 1: Simple loops on $\Sigma_{g,m}^n$ ($r=m+n$) whose Dehn twists generate the pure mapping class group
• Figure 2: The loops $\delta_k$, $\delta'_k$ and $\omega_k$ (where $\alpha_1=\delta_1=\delta_1'$)
• Figure 3: The bold loop $\gamma_{1,i}$ in the middle figure is given by $t_i^{-1}(\delta_i)$; its Dehn twist is $t_i^{-1}d_it_i$. The bold loop $\gamma_{2,i}$ in the lowest figure is given by $(t_it_{i-1}')^{-1}(\delta_i)$; its Dehn twist is ${t}_{i-1}^{\prime -1}t_i^{-1}d_it_it'_{i-1}$.
• Figure 4: A-moves at the level of ribbon braids
• Figure 5: Surfaces $\Sigma_{2,5}$ cut off from $\Sigma_{g,n}=\Sigma_{4,1}$

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 2.1
• Definition 2.2
• Theorem 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• Theorem 3.1: Wajnryb W
• proof