Classification problems in symbolic dynamics

During this talk I will discuss the descriptive set-theoretic
complexity of the topological conjugacy relation for free minimal
G-subshifts for various countable groups G. For residually finite countable
groups G we will see that there exists a probability measure on the set of
free minimal G-subshifts, which is invariant under a natural action of G
and such that the stabilizers of points in this action are a.e. amenable.
As a consequence, we will get that if G is a countable residually finite
non-amenable group, then the relation of topological conjugacy on free
minimal G-subshifts is not amenable. On the other hand, for the group G=Z,
we will look at the class of Toeplitz subshifts with separated holes and
see that the conjugacy relation is an amenable equivalence relation there.
This is joint work (in progress) with Todor Tsankov.