Subsemigroups of Amenable Groups

Subsemigroups of Cancellative Amenable Semigroups

We generalize a theorem of Frey by giving sufficient conditions for a subsemigroup T of a cancellative left amenable semigroup S to be left amenable. In particular, we show that if S is left amenable and T does not contain a free subsemigroup on two generators, then T is left amenable as well.

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 ∈ ...

Presentations for Subsemigroups of Groups

This thesis studies subsemigroups of groups from three perspectives: automatic structures, ordinary semigroup presentations, and Malcev presentaions. [A Malcev presentation is a presentation of a special type for a semigroup that can be embedded into a group. A group-embeddable semigroup is Malcev coherent if all of its finitely generated subsemigroups admit finite Malcev presentations.] The th...

Structural Properties of Inner Amenable Discrete Groups

Some extremely amenable groups

A topological group G is extremely amenable if every continuous action of G on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology but not with the uniform one. Strengthening a de la Harpe’s result, we sh...