Degree Name

Doctor of Philosophy


School of Mathematics and Applied Statistics


In this thesis, we study self-similar actions of groupoids on row-finite directed graphs and their associated Cuntz-Krieger algebras. Roughly speaking, if for all scales of parts of the object reiterate the whole, then the object is self-similar. For algebraic objects such as groups or groupoids that have this self-similarity property, we simply call them selfsimilar groups or self-similar groupoids. As an illustration of this self-similarity property, we recall the addition algorithm of integers that we learned from primary school, there is a so-called carrying operation that takes place to deal with addition of larger integers. Analogous to this carrying operation there is so-called a restriction map that encodes self-similarity property.

In the 1980s, Grigorchuk, Gupta, and Sidki were among the pioneers who introduced the concept of self-similar groups to address the question of whether there exist groups with intermediate growth. These groups exhibit self-similarity in their actions on the path-spaces of graphs with a single vertex. In contrast, Laca, Raeburn, Rammage, and Whittaker (2018) extended this notion to encompass self-similar actions on more general directed graphs by introducing the concept of a self-similar groupoid. A self-similar groupoid is defined as a system of partial isomorphisms of the path-spaces of a finite directed graph. Laca et al. (2018) explored self-similar groupoids and their associated C∗-algebras to investigate the KMS states on their corresponding dynamical systems. In contrast to their approach, which employed Hilbert modules and Cuntz-Pimsner algebras, we focus solely on generators and relations, as well as the associated Cuntz-Krieger algebras. Moreover, we extend our analysis to a broad class of self-similar groupoids, with particular emphasis on the ideal structure of their associated C∗-algebras.

FoR codes (2020)

490408 Operator algebras and functional analysis



Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.