Automorphism-preserving color substitutions on Profinite Graphs
Abstract
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. This thesis explores a different approach to study profinite groups using profinite graphs and that is with the notion of automorphisms and colors. It contains 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 establishes 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. The thesis contains an application of these theorems, which is a 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.