# Groups of smooth diffeomorphisms of Cantor sets embedded in a line

## TL;DR

This paper investigates the properties of diffeomorphism groups of Cantor sets embedded in the real line, revealing finiteness conditions and structural constraints on their subgroups, extending known results for Higman-Thompson groups.

## Contribution

It proves new properties of these diffeomorphism groups, including finiteness of finitely generated torsion subgroups and virtual abelianness of certain subgroups, generalizing prior results for Vn groups.

## Key findings

- Finitely generated torsion subgroups are finite.
- Subgroups without free subsemigroups are virtually abelian.
- Groups do not contain non-virtually abelian nilpotent subgroups.

## Abstract

Let K be a Cantor set embedded in the real line R. Following Funar and Neretin, we define the diffeomorphism group of K as the group of homeomorphisms of K which locally look like a diffeomorphism between two intervals of R. Higman-Thompson's groups Vn appear as subgroups of such groups. In this article, we prove some properties of this group. First, we study the Burnside problem in this group and we prove that any finitely generated subgroup consisting of finite order elements is finite. This property was already proved by Rover in the case of the groups Vn. We also prove that any finitely generated subgroup H without free subsemigroup on two generators is virtually abelian. The corresponding result for the groups Vn was unknown to our knowledge. As a consequence, those groups do not contain nilpotent groups which are not virtually abelian.