Scale-multiplicative semigroups and geometry: automorphism groups of trees

A scale-multiplicative semigroup in a totally disconnected, locally compact group $G$ is one for which the restriction of the scale function on $G$ is multiplicative. The maximal scale-multiplicative semigroups in groups acting 2-transitively on the set of ends of trees without leaves are determined in this paper and shown to correspond to geometric features of the tree.Comment: submitted to Groups, Geometry, and Dynamic

Equilibrium states on the Cuntz-Pimsner algebras of self-similar actions

We consider a family of Cuntz-Pimsner algebras associated to self-similar group actions, and their Toeplitz analogues. Both families carry natural dynamics implemented by automorphic actions of the real line, and we investigate the equilibrium states (the KMS states) for these dynamical systems. We find that for all inverse temperatures above a critical value, the KMS states on the Toeplitz algebra are given, in a very concrete way, by traces on the full group algebra of the group. At the critical inverse temperature, the KMS states factor through states of the Cuntz-Pimsner algebra; if the self-similar group is contracting, then the Cuntz-Pimsner algebra has only one KMS state. We apply these results to a number of examples, including the self-similar group actions associated to integer dilation matrices, and the canonical self-similar actions of the basilica group and the Grigorchuk group.Comment: The paper has been updated to agree with the published versio

On some twisted Kac-Moody groups

This thesis consists of two distinct parts. The first part comprises the first three chapters and is largely of an expository nature. The second part comprises the last three chapters all of which are, to the best of our knowledge, original. In the first part we cover the background material which we shall require in the sequel. Thus Chapter 1 deals with the theory of Kac-Moody algebras and is drawn from two main sources, namely [Kac90] and [BdK90]. Two enlightening examples are given at the end of this chapter. Chapter 2 introduces the notion of the Kac-Moody group functor. This material is drawn largely, but not exclusively, from an extensive body of work on the topic by J. Tits. We give a presentation for Kac-Moody groups over fields and describe some of their properties. In Chapter 3 we give an overview of some results on Kac-Moody groups. First we describe the work of J-Y. Hee generalizing the notion of twisted Chevalley groups to the Kac-Moody situation. We then give an exposition of the work of R.W. Carter and Y. Chen on the automorphisms of complex simply-connected affine Kac-Moody groups arising from extended Cartan matrices and we describe the classification of such automorphisms. In particular, we note that the family of diagonal automorphisms of such groups behave in a manner which has no analogy in the classical theory. We conclude the Chapter with an example demonstrating the limitation of Hee’s results with regards to this type of automorphism. Chapter 4 makes use of the results on Kac-Moody algebras described in §1.5 to extend the results of Hee. Suppose A is a simply-laced extended Cartan matrix and let β(K) be a Kac-Moody group associated to A. In Chapter 4 we extend the results of Hee to the fixed point subgroup, β(K) say, of β(K) under a particular graph-by-diagonal automorphism. We then establish an isomorphism between the subgroup β(K) so obtained and a Kac-Moody group associated to an affine Cartan matrix of type II or III. Thus Chapter 4 contains our main contributions for two reasons. Firstly, it provides a realization of Kac-Moody groups of types II and III in terms of those arising from extended Cartan matrices. More precisely, Propositions 4.4.3, 4.5.6, and 4.6.4 prove the following result