Groups acting on rooted trees and their representations on the boundary

Steffen Kionke

Published 2018-01-08Version 1

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. On one hand, this can be used to study Gelfand pairs and enables us to answer a question of Grigorchuk. On the other hand, the result can be used in the investigation of representation zeta functions. To provide examples, we analyse in detail the local 2-transitivity of GGS-groups.