Groups acting on rooted trees and their representations on the boundary

TL;DR

This paper studies how groups acting on rooted trees influence boundary representations, introducing local 2-transitivity, and provides decomposition results, examples, and applications to Gelfand pairs and zeta functions.

Contribution

It introduces the concept of local 2-transitivity for group actions on rooted trees and derives a decomposition of boundary representations under this condition.

Findings

Decomposition of boundary representations into irreducibles under local 2-transitivity

Analysis of GGS-groups for local 2-transitivity

Explicit formulas for zeta functions of induced representations

Abstract

We consider groups that act on spherically symmetric rooted trees and study the associated representation of the group on the space of locally constant functions on the boundary of the tree. We introduce and discuss the new notion of locally 2-transitive actions. Assuming local 2-transitivity our main theorem yields a precise decomposition of the boundary representation into irreducible constituents. The method can be used to study Gelfand pairs and enables us to answer a question of Grigorchuk. To provide examples, we analyse in detail the local 2-transitivity of GGS-groups. Moreover, our results can be used to determine explicit formulae for zeta functions of induced representations defined by Klopsch and the author.