Automorphism groups of finite topological rank

TL;DR

This paper provides criteria for automorphism groups of ultrahomogeneous and $mbda$-categorical structures to have finite topological rank, showing many such structures in literature share this property and exploring implications and open questions.

Contribution

It introduces a criterion for finite topological rank of automorphism groups and applies it broadly to $mbda$-categorical structures, expanding understanding of their symmetry groups.

Findings

Automorphism groups of many $mbda$-categorical structures have finite topological rank.

Conditions for automorphism groups to be topologically 2-generated and have cyclically dense conjugacy classes.

Most $mbda$-categorical structures without compact quotients have automorphism groups of finite topological rank.

Abstract

We offer a criterion for showing that the automorphism group of an ultrahomogeneous structure is topologically 2-generated and even has a cyclically dense conjugacy class. We then show how finite topological rank of the automorphism group of an $\omega$-categorical structure can go down to reducts. Together, those results prove that a large number of $\omega$-categorical structures that appear in the literature have an automorphism group of finite topological rank. In fact, we are not aware of any $\omega$-categorical structure to which they do not apply (assuming the automorphism group has no compact quotients). We end with a few questions and conjectures.