Groups of tree automorphisms as diffeological groups

TL;DR

This paper explores the structure of groups of tree automorphisms as diffeological groups, revealing that the natural diffeology on the automorphism group is discrete, which impacts understanding of their differentiable structure.

Contribution

It introduces a diffeological framework for analyzing groups of tree automorphisms and demonstrates that the automorphism group has a discrete diffeology, clarifying its differentiable structure.

Findings

The natural diffeology on the automorphism group is discrete.

Diffeological structures on regular trees can be aligned with the standard topology.

Subgroups of automorphisms also inherit the discrete diffeology.

Abstract

We consider certain groups of tree automorphisms as so-called diffeological groups. The notion of diffeology, due to Souriau, allows to endow non-manifold topological spaces, such as regular trees that we look at, with a kind of a differentiable structure that in many ways is close to that of a smooth manifold; a suitable notion of a diffeological group follows. We first study the question of what kind of a diffeological structure is the most natural to put on a regular tree in a way that the underlying topology be the standard one of the tree. We then proceed to consider the group of all automorphisms of the tree as a diffeological space, with respect to the functional diffeology, showing that this diffeology is actually the discrete one, the fact that therefore is true for its subgroups as well.