Rigidity of braid group actions on $\mathbb{R}$ and of low-genus mapping class group actions on $S^1$
[Rigidité des actions du groupe des tresses sur $\mathbb{R}$ et des actions du groupe modulaire de surface de faible genre sur $S^1$]
Ba, Idrissa1  ; Clay, Adam1  ; Ghaswala, Tyrone2
1Department of Mathematics, University of Manitoba, Winnipeg, MB Canada R3T 2N2
2Center for Education in Mathematics, and Computing, University of Waterloo, Waterloo, ON Canada N2L 3G1
Annales de l'Institut Fourier, Online first, 33 p.

Every nontrivial action of the braid group $B_n$ on $\mathbb{R}$ by orientation-preserving homeomorphisms yields, up to conjugation by a homeomorphism of $\mathbb{R}$, a representation $\rho : B_n \rightarrow \operatorname{H\widetilde{ome}o}_+(S^1)$ and therefore determines a translation number for every element of $B_n$. In this manuscript we offer a simple characterisation of which actions of $B_n$ on $\mathbb{R}$ produce translation numbers that agree with those arising from the standard Nielsen–Thurston action on $\mathbb{R}$. Our approach is to prove an analogous statement concerning left orderings of $B_n$ via a technique that uses the space of left orderings of $B_n$, the isolated points in this space, and the natural conjugacy action of $B_n$. We use this result to extend recent rigidity results of Mann and Wolff concerning mapping class group actions on $S^1$ to the case of low-genus surfaces with marked points.

Toute action non triviale du groupe des tresses $B_n$ sur $\mathbb{R}$ par des homéomorphismes préservant l’orientation conduit, à conjugaison près par un homéomorphisme de $\mathbb{R}$, à une représentation $\rho : B_n \rightarrow \mathrm{H\widetilde{ome}o}_+(S^1)$ et détermine ainsi un nombre de translation pour chaque élément de $B_n$. Dans ce manuscrit, nous proposons une caractérisation simple des actions de $B_n$ sur $\mathbb{R}$ produisant des nombres de translation qui concordent avec ceux issus de l’action standard de Nielsen–Thurston sur $\mathbb{R}$. Notre approche consiste à démontrer un énoncé analogue concernant les ordres à gauche sur $B_n$ à l’aide d’une technique utilisant l’espace des ordres à gauche sur $B_n$, les points isolés dans cet espace, et l’action naturelle de $B_n$ par conjugaison. Nous utilisons ce résultat pour étendre les résultats récents de rigidité de Mann et Wolff sur les actions des groupes modulaires de surfaces sur $S^1$ au cas des surfaces de faible genre avec points marqués.

Reçu le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3788
Classification : 20F36, 06F15, 20F60
Keywords: Braid groups, mapping class groups, ordered groups, actions on $\mathbb{R}$ and $S^1$
Mots-clés : Groupes des tresses, groupe modulaire de surface, groupes ordonnés, actions sur $\mathbb{R}$ et $S^1$.

Ba, Idrissa  1   ; Clay, Adam  1   ; Ghaswala, Tyrone  2

1 Department of Mathematics, University of Manitoba, Winnipeg, MB Canada R3T 2N2
2 Center for Education in Mathematics, and Computing, University of Waterloo, Waterloo, ON Canada N2L 3G1 
Ba, Idrissa; Clay, Adam; Ghaswala, Tyrone. Rigidity of braid group actions on $\mathbb{R}$ and of low-genus mapping class group actions on $S^1$. Annales de l'Institut Fourier, Online first, 33 p.
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