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.