On the group of spheromorphisms of a homogeneous non-locally finite treeстатья
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Аннотация:We consider a tree $T$ all whose vertices have countable valency. Its boundary is the Baire space $B={\mathbb N}^{\mathbb N}$ and the set of irrational numbers ${\mathbb R}\setminus {\mathbb Q}$ is identified with $B$ by continued fraction expansions. Removing $k$ edges from $T$, we get a forest consisting of copies of $T$. A spheromorphism (or hierarchomorphism) of $T$ is an isomorphism of two such subforests regarded as a transformation of $T$ or $B$. We denote the group of all spheromorphisms by $Hier(T)$. We show that the correspondence ${\mathbb R}\setminus {\mathbb Q}≃B$ sends the Thompson group realized by piecewise $PSL_2({\mathbb Z})-transformations to a subgroup of $Hier(T)$. We construct some unitary representations of $Hier(T)$, show that the group $Aut(T)$ of automorphisms is spherical in $Hier(T)$ and describe the train (enveloping category) of $Hier(T)$.