Elementary topological groups

Topological groups are central in the study of many objects in mathematics. The group structure facilitates the description of symmetries inherent in these objects, while the associated topology provides tools for understanding their interactions. For instance, in the study of Polish groups, a countable basis for the topology proves to be a valuable tool for understanding it, even when the group itself is uncountable. When dealing with totally disconnected, compactly generated groups, one can use their Cayley-Abels graph. This discrete, countable structure can be used to describe the group, revealing properties intricately connected to its overall characteristics. Two main classes of topological groups are connected and totally disconnected groups. In this thesis, we focus on the latter, more specifically on a subclass of it, the totally disconnected, locally compact elementary groups. Such groups play an important role in understanding the general structure of totally disconnected, locally compact groups, for example, in classifying the chief factors of totally disconnected groups. In Chapter 3, our attention is directed toward the decomposition rank — a rank defined for the class of elementary totally disconnected, locally compact groups that measure their complexity. It is not known whether or not such rank has a countable upper bound, and such a question also relates to other problems in the theory of totally disconnected, locally compact groups. We show the existence of elementary groups with rank up to $omega^omega+1$ with a chief factor of a special type. We also introduce the concept of the residual height, using it to build explicit examples of elementary groups with rank equal to any given successor ordinal up to $omega^omega+1$. Chapter 4 is divided into two topics we worked with during my PhD research. Firstly, we investigate the concept of a rank for totally disconnected, locally compact groups, associated with the relative Tits core of their elements, which we refer to as the rTc rank. We establish connections between the rTc rank and the previously discussed decomposition rank. In the second part, we introduce two distinct classes of subgroups that densely embed into elementary totally disconnected, locally compact groups. We utilize these classes to examine algebraic properties that serve as indicators for a topological group to be classified as an elementary totally disconnected, locally compact group. In Chapter 5, we focus on the study of incidence rings and their groups of units from a topological perspective. Our work includes proving the isomorphism problem for incidence rings over partially ordered sets through topological arguments. We also utilize category theory to establish connections between the categories of incidence rings and the class of preordered sets. We also extend Peter Groenhout's construction of simple topological infinite matrix groups to encompass a broader class of groups. We conclude the chapter by illustrating that elementary groups with groups of units from specific incidence rings as open subgroups have a decomposition rank bounded above by $omega$.