Sofic representations of amenable groups

A Sofic Group Away from Amenable Groups

We give an example of a sofic group, which is not a limit of amenable groups.

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

Sofic groups and convolution operators

Conjecture: Let Γ be a discrete group, f ∈ L(Γ) and Tf : L (Γ) → L(Γ) be the right convolution by f . Then if Tf is injective, it is surjective as well. We prove the conjecture for the class of sofic groups. AMS Subject Classification: 43A20

Sofic Groups and Diophantine Approximation

We prove the algebraic eigenvalue conjecture of J.Dodziuk, P.Linnell, V.Mathai, T.Schick and S.Yates (see [2]) for sofic groups. Moreover, we give restrictions on the spectral measure of elements in the integral group ring. Finally, we prove a quantization of the operator norm below 2. To the knowledge of the author, there is no group known, which is not sofic.