Extensions of amenable groups by recurrent groupoids

Extensions of amenable groups by recurrent groupoids

We show that amenability of a group acting by homeomorphisms can be deduced from a certain local property of the action and recurrency of the orbital Schreier graphs. This applies to a wide class of groups, amenability of which was an open problem, as well as unifies many known examples to one general proof. In particular, this includes Grigorchuk’s group, Basilica group, groups associated with...

Extensions of Positive Definite Functions on Amenable Groups

Let S be a subset of a amenable group G such that e ∈ S and S = S. The main result of the paper states that if the Cayley graph of G with respect to S has a certain combinatorial property, then every positive definite operator-valued function on S can be extended to a positive definite function on G. Several known extension results are obtained as a corollary. New applications are also presented.

Recurrent Points and Discrete Points for Elementary Amenable Groups

Let fig be the Stone-Cech compactification of a group G, Aa the set of all almost periodic points in G, Ka c[U { supp eLIM(G)}] and Ra the set of all recurrent points in fiG. In this paper we will study the relationships between Ka and Ra, and between Aa and Ra. We will show that for any infinite elementary amenable group G, Aa Ra and RaKa =/= .

Amenable Groups

Throughout we let Γ be a discrete group. For f : Γ → C and each s ∈ Γ we define the left translation action by (s.f)(t) = f(s−1t). Definition 1.1. A group Γ is amenable is there exists a state μ on l∞(Γ) which is invariant under the left translation action: for all s ∈ Γ and f ∈ l∞(Γ), μ(s.f) = μ(f). Example 1.2. Finite groups are amenable: take the state which sends χ{s} to 1 |Γ| for each s ∈ ...

Structural Properties of Inner Amenable Discrete Groups