I have introduced Polish structures in order to apply model theoretic ideas
in the studies of purely descriptive set theoretic and topological objects
such as Polish $G$-spaces or, more generally, Borel $G$-spaces. In
particular, Polish structures generalize profinite structures introduced by
Newelski. Polish structures allow us to apply ideas and techniques from
model theory, descriptive set theory, topology and the theory of profinite
groups.
A Polish structure is a pair $(X,G)$ where $G$ is a Polish group acting
(faithfully) on a set $X$ so that the stabilizers of all points are closed
subgroups of $G$. We say that $(X,G)$ is small if for every natural number
$n$ there are countably many orbits on $X^n$ under $G$.
A simple non-profinite example of a small Polish structure is the unit
circle with the full group of homeomorphisms. In fact, most natural examples
of compact metric spaces with the full group of homeomorphisms are small
Polish structures. More complicated examples are Hilbert cube and the
pseudo-arc with the full group of homeomorphisms.
I will discuss a purely topological notion of independence, called
non-meager independence, that satisfies some nice properties (e.g. symmetry,
transitivity, existence of independent extensions) in small Polish
structures, and so allows us to introduce basic stability-theoretic concepts
and to prove fundamental results about them (e.g. Lascar inequalities). This
notion of independence coincides with $m$-independence introduced by
Newelski in profinite structures.
In the second part of my talk I will concentrate on the structure of small
compact $G$-groups, i.e. small Polish structures $(X,G)$ where $X$ is a
compact group and $G$ acts continuously on $X$ as a group of automorphisms.
I will present an example of such a group which is not solvable-by-finite.
On the other hand, under a natural model theoretic assumption of
'superstability' with respect to $nm$-independence, each such group is
solvable-by-finite, and assuming finiteness of the underlying rank, it is
even nilpotent-by-finite.
I will finish with some open questions.