Amenability and profinite completions of finitely generated groups

Abstract: In this talk we explore the interplay between the finite quotients of finitely generated residually finite groups and the concept of amenability. We will show that the question whether a residually finite group is amenable cannot be decided by looking at its finite quotients.

More precisely, we construct a finitely generated, residually finite, amenable group $A$ and an uncountable family of finitely generated, residually finite non-amenable groups all of which are profinitely isomorphic to $A$. All of these groups are branch groups. Moreover, picking up Grothendieck's problem, the group $A$ embeds in these groups such that the inclusion induces an isomorphism of profinite completions.

In addition, we review the concept of uniform amenability, a strengthening of amenability introduced in the 70's, and we prove that uniform amenability indeed is detectable from the profinite completion.

This is based on joint work with Steffen Kionke.