Automorphism groups of countable structures and groups of measurable functions

TL;DR

This paper investigates the properties of automorphism groups of countable structures and groups of measurable functions, establishing conditions under which these groups have ample generics and analyzing their topological similarity and conjugacy classes.

Contribution

It proves that for Polish groups, having ample generics in the measurable function group implies the original group has ample generics, and it characterizes the topological similarity classes of automorphism groups of countable structures.

Findings

If $L_0(G)$ has ample generics, then $G$ has ample generics.

For automorphism groups of countable structures, similarity classes are either precompact, discrete, or meager.

A trichotomy for the structure of similarity and conjugacy classes in automorphism groups and their measurable function groups.

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 $L_0({\rm{Aut}}(M))$.