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

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 $\mathrm{Hier}(\mathbb{T})$.

arXiv comment37p., 9 fig