Full groups of minimal homeomorphisms

Tomás Ibarlucia (1) , Julien Melleray (1)

1 ICJ - Institut Camille Jordan

We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space $X$, showing that these groups do not admit a compatible Polish group topology and, in the case of $\Z$-actions, are coanalytic non-Borel inside $\Homeo(X)$. We then focus on the closure of the full group of a uniquely ergodic homeomorphism, elucidating under which conditions this group has a comeager (or, equivalently in that case, dense) conjugacy class, and point out that when that happens the closure of the full group is simple and satisfies the automatic continuity property. We also prove that the full group of a uniquely ergodic homeomorphism is topologically simple.

Domaines: Logique [math.LO]; Systèmes dynamiques [math.DS]

Tomás Ibarlucia, Julien Melleray. Full groups of minimal homeomorphisms. 2013. ⟨hal-00861053v1⟩