PhD Thesis Defence Public Lecture (Math) - Michal Cizek

Profinite groups are topological groups which are known to be Galois groups. Their free product was extensively studied by Luis Ribes and Pavel Zaleskii using the notion of a profinite graph and having profinite groups act freely on such graphs.

We will explore a different approach to study profinite groups with profinite graphs using the notion of automorphisms and colors. We will go over a generalization to profinite graphs of the theorem of Frucht (1939) that shows that every finite group is a group of automorphisms of a finite connected graph, and establish a profinite analog of the theorem of Sabidussi (1959) that states that every abstract group is a group of automorphisms of a connected graph.

The profinite version of these theorems is: Every finitely generated profinite group is a group of continuous automorphisms of a profinite graph with a closed set of edges and every profinite group is a group of continuous automorphisms of a connected profinite graph.

We will finally go over an application of these theorems, which is a proof of the conjecture of Sidney Morris and Karl Hoffmann stating that every profinite group is a group of autohomeomorphisms of a connected compact Hausdorff space.