Consider an infinite homogeneous tree of valence , its group Aut of automorphisms, and the group Hier of its spheromorphisms (hierarchomorphisms), i.e., the group of homeomorphisms of the boundary of that locally coincide with transformations defined by automorphisms. We show that the subgroup Aut is spherical in Hier, i.e., any irreducible unitary representation of Hier contains at most one Aut-fixed vector. We present a combinatorial description of the space of double cosets of Hier with respect to Aut and construct a “new” family of spherical representations of Hier. We also show that the Thompson group Th has PSL-spherical unitary representations.
Cite this article
Yury A. Neretin, On spherical unitary representations of groups of spheromorphisms of Bruhat–Tits trees. Groups Geom. Dyn. 15 (2021), no. 3, pp. 801–824DOI 10.4171/GGD/612