On the unitary representation theory of contraction groups

The study of unitary representations of locally compact groups is a classical area of investigation in mathematics, with deep roots in early 20th century quantum physics. Connected locally compact groups, and in particular connected Lie groups, have been the primary focus of unitary representation theory over the past century. Representation theoretic techniques have also played a fundamental role in understanding the structure of these groups. Totally disconnected locally compact groups (tdlc groups), on the other hand, are not as well understood as their connected counterparts, and modern research in locally compact group theory is focussed on understanding the structure of these groups. Automorphism groups of trees play an important role in theory of tdlc groups, and they have been referred to as a "microcosm" for the overarching theory. The unitary representation theory of tdlc groups, and in particular, automorphism groups of trees, is not well understood in contrast to that of Lie groups, for example. Recent research has provided greater understanding of the unitary representation theory of non-amenable automorphism groups of trees, however, the unitary representation theory of amenable automorphism groups of trees has gone relatively uninvestigated. This thesis focusses on understanding the unitary representation theory of certain classes of amenable automorphism groups of trees: we find new classes of type I and non-type-I amenable automorphism groups of trees and classify the irreducible unitary representations of the type I groups. In doing this, we explore novel connections between the classical unitary representation theory of Lie groups and algebraic groups, and the modern theory of tdlc groups.