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Infinite Dimensional Spherical Analysis and Harmonic Analysis for Groups acting on Homogeneous Trees

by Emil Axelgaard
PhD Dissertations | August 2012
In this thesis, we study groups of automorphisms for homogeneous trees of countable degree by using an inductive limit approach. The main focus is the thourough discussion of two Olshanski spherical pairs consisting of automorphism groups for a homogeneous tree and a homogeneous rooted tree, respectively. We determine the spherical functions, discuss their positive definiteness, and make realizations of the corresponding spherical representations. We turn certain double coset spaces into semigroups and use this to make a complete classification of a certain class of unitary representations of the groups, the so-called irreducible tame representations. We prove the existence of irreducible non-tame representations by constructing a compactification of the boundary of the tree - an object which until now has not played any role in the analysis of automorphism groups for trees which are not locally finite. Finally, we discuss conditionally positive definite functions on the groups and use the generalized Bochner-Godement theorem for Olshanski spherical pairs to prove Levy-Khinchine formulas for both of the considered pairs.
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Thesis advisor: Bent Ørsted