Distortion and Tits alternative in smooth mapping class groups

TL;DR

This paper investigates the structure of smooth mapping class groups of surfaces relative to a Cantor set, demonstrating the absence of distorted elements in certain subgroups and establishing a weak Tits alternative.

Contribution

It provides new results on the algebraic structure of these groups, specifically regarding distortion and Tits alternative properties.

Findings

Subgroup of diffeomorphisms isotopic to identity contains no distorted elements.

Established a weak Tits alternative for these groups.

Results apply to the standard ternary Cantor set.

Abstract

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.