Automorphism groups of countable structures and groups of measurable functions

Abstract

Let $G$ be a topological group and let $\mu$ be the Lebesgue measure on the interval $[0,1]$. We let $L_0(G)$ to be the topological group of all $\mu$-equivalence classes of $\mu$-measurable functions defined on [0,1] with values in $G$, taken with the pointwise multiplication and the topology of convergence in measure. We show that for a Polish group $G$, if $L_0(G)$ has ample generics, then $G$ has ample generics, thus the converse to a result of Ka\"{i}chouh and Le Ma\^{i}tre. We further study topological similarity classes and conjugacy classes for many groups ${\rm{Aut}}(M)$ and $L_0({\rm{Aut}}(M))$, where $M$ is a countable structure. We make a connection between the structure of groups generated by tuples, the Hrushovski property, and the structure of their topological similarity classes. In particular, we prove the trichotomy that for every tuple $ \bar{f}$ of ${\rm{Aut}}(M)$, where $M$ is a countable structure such that algebraic closures of finite sets are finite, either the countable group $\langle \bar{f} \rangle$ is precompact, or it is discrete, or the similarity class of $\bar{f}$ is meager, in particular the conjugacy class of $\bar{f}$ is meager. We prove an analogous trichotomy for groups $L_0({\rm{Aut}}(M))$.