Distortion and Tits alternative in smooth mapping class groups

Sebastian Hurtado (1) , Emmanuel Militon (2)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 LJAD - Laboratoire Jean Alexandre Dieudonné

In this article, we study the smooth mapping class group of a surface S relative to a given Cantor set, that is the group of isotopy classes of orientation-preserving smooth diffeomorphisms of S which preserve this Cantor set. When the Cantor set is the standard ternary Cantor set, we prove that the subgroup consisting of diffeomorphisms which are isotopic to the identity on S does not contain any distorted elements. Moreover, we prove a weak Tits alternative for these groups.

Domaines: Systèmes dynamiques [math.DS]; Théorie des groupes [math.GR]; Topologie géométrique [math.GT]

Sebastian Hurtado, Emmanuel Militon. Distortion and Tits alternative in smooth mapping class groups. Transactions of the American Mathematical Society, 2019, 371 (12), pp.8587-8623. ⟨hal-01161004v4⟩