Skew-product graph of groups and their toolkits

Abstract

The K-theory associated with a C*-algebra plays a fundamental role in the classification and structural understanding of C*-algebras.

This thesis investigates the C*-algebras associated with graphs of groups, a rich mathematical structure first systematically developed by Bass and Serre in their foundational work on group actions on trees. We adapt and extend the skew-product construction for directed graphs to the graph of groups setting. Specifically, given a cocycle labelling the edges of a graph of groups by a discrete group, a definition of skew-product graphs of groups is provided. The main theoretical contribution demonstrates that there is a natural connection between the skew-product graph of groups C*-algebra and the crossed product by the induced coaction. In addition, this definition of skew-product graphs of groups is shown to be consistent with the existing definition of skew-product graphs, in terms of the directed graph associated to graphs of groups E_G. Finally, using the existing isomorphism between graphs of groups C*-algebras and its fibred product groupoid algebra, the isomorphism between the skew-product graph of groups C*-algebra and the crossed product by the induced coaction is extended to the crossed product fibred product groupoid algebra by coaction.

This thesis also includes a survey of K-theory for C*-algebras, including the K-theory of graph algebras. The last chapter also acts as a literature review of the recent developments in K-theory for both graph of groups C*-algebras and graph of groups actions on multitrees.