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

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 $\mathbb B$\mathrm{.}Denotethegroupofallspheromorphismsby$$\mathrm{H}\mathrm{i}\mathrm{e}\mathrm{r}$({\mathbb T})$. We a show that the correspondence ${\mathbb R}\setminus{\mathbb Q}\simeq{\mathbb B}$sendstheThompsongrouprealizedbypiecewise$$\mathrm{P}\mathrm{S}\mathrm{L}$_2({\mathbb Z})$-transformationstoasubgroupof$$\mathrm{H}\mathrm{i}\mathrm{e}\mathrm{r}$({\mathbb T})$. We construct some unitary representations of the group $$\mathrm{H}\mathrm{i}\mathrm{e}\mathrm{r}$({\mathbb T})$,showthatthegroupofautomorphisms$$\mathrm{A}\mathrm{u}\mathrm{t}$({\mathbb T})$ is spherical in $\mathrm{H}\mathrm{i}\mathrm{e}\mathrm{r}(\mathbb{T})$, and describe the train (enveloping category) of $\mathrm{H}\mathrm{i}\mathrm{e}\mathrm{r}(\mathbb{T})$.