A positive proportion of elements of mapping class groups is pseudo-Anosov

María Cumplido Cabello (1) , Bert Wiest (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero. We also prove similar results for a large class of subgroups of the mapping class group.

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

María Cumplido Cabello, Bert Wiest. A positive proportion of elements of mapping class groups is pseudo-Anosov. Bulletin of the London Mathematical Society, 2018, 50 (3), pp.390-394. ⟨10.1112/blms.12148⟩. ⟨hal-01497687⟩