On the simplicity of homeomorphism groups of a tilable lamination

TL;DR

This paper proves the simplicity of the identity component of leaf-preserving homeomorphism groups in R^d-tilable laminations and shows uniform perfectness in one-dimensional cases, extending to dense subgroups.

Contribution

It establishes the simplicity and uniform perfectness of certain homeomorphism groups associated with tilable laminations, advancing understanding of their algebraic structure.

Findings

The identity component of leaf-preserving homeomorphisms is simple.

In one dimension, this group is uniformly perfect.

Similar results hold for dense subgroups of these homeomorphisms.

Abstract

We show that the identity component of the group of homeomorphisms that preserve all leaves of a R^d-tilable lamination is simple. Moreover, in the one dimensional case, we show that this group is uniformly perfect. We obtain a similar result for a dense subgroup of homeomorphisms.