On the group of spheromorphisms of the homogeneous non-locally finite tree

TL;DR

This paper studies the group of spheromorphisms of a non-locally finite homogeneous tree, linking it to the Thompson group, constructing representations, and analyzing its automorphisms and categorical structure.

Contribution

It introduces the group of spheromorphisms for such trees, connects it to the Thompson group via boundary identification, and explores its representations and automorphism properties.

Findings

Thompson group embeds into spheromorphisms group

Constructed unitary representations of the spheromorphisms group

Described the automorphism group and categorical structure

Abstract

Consider a tree $\mathbb T$, all whose vertices have countable valence; its boundary is the Baire space $\mathbb{B} \simeq\mathbb{N}^{\mathbb N}$; continued fractions expansions identify the set of irrational numbers $\mathbb{R}\setminus \mathbb{Q}$ with $\mathbb B$. Removing $k$ edges from $\mathbb T$ we get a forest consisting of copies of $\mathbb T$. A spheromorphism (or hierarchomorphism) of $\mathbb T$ is an isomorphisms of two such subforests considered as a transformation of $\mathbb T$ or of $\mathbb B$. Denote the group of all spheromorphisms by $\mathrm{Hier}({\mathbb T})$. We a show that the correspondence ${\mathbb R}\setminus{\mathbb Q}\simeq{\mathbb B}$ sends the Thompson group realized by piecewise $\mathrm{PSL}_2({\mathbb Z})$-transformations to a subgroup of $\mathrm{Hier}({\mathbb T})$. We construct some unitary representations of the group $\mathrm{Hier}({\mathbb T})$, show that the group of automorphisms $\mathrm{Aut}({\mathbb T})$ is spherical in $\mathrm{Hier}({\mathbb T})$, and describe the train (enveloping category) of $\mathrm{Hier}({\mathbb T})$.