Diffeomorphisms groups of tame Cantor sets and Thompson-like groups

TL;DR

This paper explores the structure of diffeomorphism groups of sparse Cantor sets, showing they are countable and discrete, and connects these groups to Thompson-like groups, including higher-dimensional and braided variants.

Contribution

It establishes the discreteness and countability of diffeomorphism groups for sparse Cantor sets and introduces new Thompson-like groups from these constructions.

Findings

Diffeomorphism groups of sparse Cantor sets are countable and discrete.

Thompson's groups emerge naturally from central ternary Cantor sets.

Higher-dimensional and braided Thompson groups are constructed from Cantor set products.

Abstract

The group of $\mathcal C^1$-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson's groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin's higher dimensional generalizations $nV$ of Thompson's group $V$ arise when we consider products of central ternary Cantor sets. We derive that the $\mathcal C^2$-smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.