CAT(0) cube complexes and inner amenability

Abstract

We here consider inner amenability from a geometric and group theoretical perspective. We prove that for every non-elementary action of a group G on a finite dimensional irreducible CAT(0) cube complex, there is a nonempty G-invariant closed convex subset such that every conjugation invariant mean on G gives full measure to the stabilizer of each point of this subset. Specializing our result to trees leads to a complete characterization of inner amenability for HNN-extensions and amalgamated free products. One novelty of the proof is that it makes use of the existence of certain idempotent conjugation-invariant means on G.We additionally obtain a complete characterization of inner amenability for permutational wreath product groups. One of the main ingredients used for this is a general lemma which we call the location lemma, which allows us to "locate" conjugation invariant means on a group G relative to a given normal subgroup N of G. We give several further applications of the location lemma beyond the aforementioned characterization of inner amenable wreath products.

Exactness vs C*-exactness for certain non-discrete groups

“…To produce a non-exact example, simply let H be any non-exact discrete group and let H act on R 2 trivially. Then the semi-direct product R 2 ⋊ (F 6 * H) is non-exact, and it is also not inner amenable since by [7,Theorem 1.1] the only conjugation invariant mean on F 6 * H is evaluation at the identity. We will show that certain Burger-Mozes groups [5] in Aut(T d ) are too geometrically dense for our results of section 3 to apply.…”

On algebras associated with invariant means on the subnormal subgroups of an amenable group

“…Our initial motivation for studying A sn (G) was to study rad ℓ 1 (G) ′′ , and we subsequently discovered the striking way in which A sn (G) reflects the subnormal structure of G, which we believe makes it of independent interest. Moreover, Duchesne, Tucker-Drob, and Wesolek [5] have recently used the first Arens product (which they call convolution) with conjugationinvariant means in a proof which characterizes inner amenability of amalgamated free products and HNN-extensions. This suggests to us that a better understanding of the algebraic properties of objects related to invariant means could have applications to group theory further down the line.…”