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

We consider a tree all whose vertices have countable valency. Its boundary is the Baire space and the set of irrational numbers is identified with by continued fraction expansions. Removing edges from , we get a forest consisting of copies of . A spheromorphism (or hierarchomorphism) of is an isomorphism of two such subforests regarded as a transformation of or . We denote the group of all spheromorphisms by . We show that the correspondence sends the Thompson group realized by piecewise -transformations to a subgroup of . We construct some unitary representations of , show that the group of automorphisms is spherical in and describe the train (enveloping category) of .