PhD 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 use a different approach to study profinite groups using profinite graphs and that is with the notion of automorphisms and colors. We will generalize to profinite graphs: the theorem of Frucht (1939) that shows that every finite group is a group of automorphisms of a finite connected graph, and we will give a profinite analog of the theorem of Sabidussi (1959) that states that every abstract group is a group of automorphisms of a connected graph.
Our version of those theorems will be: 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.
Finally, we will give an application of these theorems, which is a partial solution to the conjecture of Sidney Morris and Karl Hoffmann stating that every profinite group is a group of autohomeomorphisms of a connected compact Hausdorff space. We will show this to be true for finitely generated profinite groups.