CAT(0) cube complexes and inner amenability

  • Bruno Duchesne

    Université de Lorraine, Nancy, France
  • Robin D. Tucker-Drob

    Texas A&M University, College Station, USA
  • Phillip Wesolek

    Zendesk, USA
CAT(0) cube complexes and inner amenability cover
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We here consider inner amenability from a geometric and group theoretical perspective. We prove that for every non-elementary action of a group on a finite dimensional irreducible CAT(0) cube complex, there is a nonempty -invariant closed convex subset such that every conjugation invariant mean on 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 .

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 relative to a given normal subgroup of . We give several further applications of the location lemma beyond the aforementioned characterization of inner amenable wreath products.

Cite this article

Bruno Duchesne, Robin D. Tucker-Drob, Phillip Wesolek, CAT(0) cube complexes and inner amenability. Groups Geom. Dyn. 15 (2021), no. 2, pp. 371–411

DOI 10.4171/GGD/601