Profinite graphs and their automorphisms

There is an important interplay between group theory and graph theory. Using notions such as covering spaces or free actions of groups on graphs, one can describe group theoretical notions such as a free and amalgamated product using graph structures. That is the foundation for Bass-Serre theory, developed in the 1970s. While the covering spaces of graphs are a great tool for describing abstract groups, they are not so effective for Galois/Profinite groups, which are topological by their nature. Thus, to apply the ideas from Bass-Serre theory to profinite groups, we require graphs that are topological as well. Such graphs are called profinite graphs and were extensively studied by Ribes and Zaleskii.

The goal of this talk is to, in the first part, introduce profinite graphs and basic notions on them such as their topology, connectivity, and automorphisms. In the second part, we will focus on one application of profinite graphs, which shows that profinite groups can be represented as certain continuous bijections over a compact space, solving a conjecture made by Sidney Morris and Karl Hoffmann.

This lecture is meant for a broad audience. Its goal is to introduce the idea of adding a profinite structure on graphs and will provide some informal motivation of key notions. My research has been supported by the University of Western Ontario and Western Academy for Advanced Research.