Dense free subgroups of automorphism groups of homogeneous partially ordered sets

TL;DR

This paper proves that automorphism groups of countable ultrahomogeneous posets contain dense free subgroups of two generators and characterizes when certain elements are comeager, expanding understanding of their topological structure.

Contribution

It demonstrates the existence of dense free subgroups in automorphism groups of ultrahomogeneous posets and characterizes their comeager cyclically dense elements.

Findings

Automorphism groups contain dense free subgroups of two generators.

Characterization of when cyclically dense elements form comeager sets.

Utilization of Schmerl's classification of ultrahomogeneous posets.

Abstract

A countable poset is ultrahomogeneous if every isomorphism between its finite subposets can be extended to an automorphism. The groups $\operatorname{Aut}(A)$ of such posets $A$ have a natural topology in which $\operatorname{Aut}(A)$ are Polish topological groups. We consider the problem whether $\operatorname{Aut}(A)$ contains a dense free subgroup of two generators. We show that if $A$ is ultrahomogeneous, then $\operatorname{Aut}(A)$ contains such subgroup. Moreover, we characterize whose countable ultrahomogeneous posets $A$ such that for each natural $m$, the set of all cyclically dense elements $\bar{g}\in\operatorname{Aut}(A)^m$ for the diagonal action is comeager in $\operatorname{Aut}(A)^m$. In our considerations we strongly use the result of Schmerl which says that there are essentially four types of countably infinite ultrahomogeneous posets.