# Groups, Geometry, and Dynamics

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**Published online first: 2021-07-23**

**DOI: 10.4171/GGD/612**

On spherical unitary representations of groups of spheromorphisms of Bruhat–Tits trees

Yury A. Neretin^{[1]}(1) University of Vienna, Austria

Consider an infinite homogeneous tree $\mathcal{T}_n$ of valence $n+1$, its group Aut$(\mathcal{T}_n)$ of automorphisms, and the group Hier$(\mathcal{T}_n)$ of its spheromorphisms (hierarchomorphisms), i.e., the group of homeomorphisms of the boundary of $\mathcal{T}_n$ that locally coincide with transformations defined by automorphisms. We show that the subgroup Aut$(\mathcal{T}_n)$ is spherical in Hier$(\mathcal{T}_n)$, i.e., any irreducible unitary representation of Hier$(\mathcal{T}_n)$ contains at most one Aut$(\mathcal{T}_n)$-fixed vector. We present a combinatorial description of the space of double cosets of Hier$(\mathcal{T}_n)$ with respect to Aut$(\mathcal{T}_n)$ and construct a “new” family of spherical representations of Hier$(\mathcal{T}_n)$. We also show that the Thompson group Th has PSL$(2,\mathbb{Z})$-spherical unitary representations.

*Keywords: *Spheromorphism, hierarchomorphism, spherical representation, spherical subgroup, infinite symmetric group, Bruhat–Tits tree

Neretin Yury: On spherical unitary representations of groups of spheromorphisms of Bruhat–Tits trees. *Groups Geom. Dyn.* Electronically published on July 23, 2021. doi: 10.4171/GGD/612 (to appear in print)