Dense free subgroups of automorphism groups of homogeneous partially ordered sets

Abstract

Abstract A countable poset is ultrahomogeneous if every isomorphism between its finite subposets can be extended to an automorphism. If A is such a poset, then the group Aut ⁡ ( A ) {\operatorname{Aut}(A)} has a natural topology in which Aut ⁡ ( A ) {\operatorname{Aut}(A)} is a Polish topological group. We consider the problem of whether Aut ⁡ ( A ) {\operatorname{Aut}(A)} contains a dense free subgroup of two generators. We show that if A is ultrahomogeneous, then Aut ⁡ ( A ) {\operatorname{Aut}(A)} contains such a subgroup. Moreover, we characterize those countable ultrahomogeneous posets A such that for each natural number m the set of all cyclically dense elements g ¯ ∈ Aut ( A ) m {\overline{g}\in\operatorname{Aut}(A)^{m}} for the diagonal action is comeager in Aut ( A ) m {\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.