Automorphism groups of countable structures and groups of measurable functions

Abstract

Let G be a topological group and let μ be the Lebesgue measure on the interval [0, 1]. We let L₀(G) be the topological group of all μ-equivalence classes of μ-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 L₀(G) has ample generics, then G has ample generics, thus the converse to a result of Kaïchouh and Le Maître.

We further study topological similarity classes and conjugacy classes for many groups Aut(M) and L₀(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 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 L₀(Aut(M)).