Monograph

#### Degree

Doctor of Philosophy

Mathematics

Minac Jan

#### 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.

#### Summary for Lay Audience

Mathematicians often try to find links between seemingly unrelated topics. This can help in solving difficult problems. For example a problem in one theory, like geometry, could be very difficult to solve by itself, but if one looks at it from the point of view of algebra, it suddenly becomes much easier. The goal of this thesis is to study relations between an old theory invented to solve equations: Galois theory and a much more recent theory that is used to represent connections: graph theory. At first glance the two topics are seemingly unrelated: on one hand we get equations like x5 − x3 + 12x + 1 = x2 − 2 and on the other we get a list of points (vertices) and connections between them. The key that binds them together is the notion of automorphism group, which one can think of as a list of swaps that preserve certain properties. In the case of equations, we swap their solutions, in the case of graphs, we swap the vertices in such a way that, after the swaps are done, the connections remain the same. In this thesis I explore the different ways in which such swaps on solutions of equations (Galois theory) can be represented as swaps on vertices (Graph theory).