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