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.

中文翻译：

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CAT(0) 立方体复合体和内部适应性

我们在这里从几何和群体理论的角度考虑内部服从性。我们证明，对于有限维不可约 CAT(0) 立方体复形上的群 $G$ 的每个非基本动作，存在一个非空的 $G$-不变闭凸子集，使得 $G$ 上的每个共轭不变均值给出对这个子集的每个点的稳定器进行全面测量。将我们的结果专门用于树导致对 HNN 扩展和合并自由产品的内部适应性的完整表征。该证明的一个新颖之处在于它利用了 $G$ 上某些幂等共轭不变手段的存在。

我们还获得了置换花圈产品组的内部适应性的完整特征。用于此的主要成分之一是我们称为位置引理的一般引理，它允许我们“定位”组 $G$ 上的共轭不变均值相对于 $G$ 的给定正常子组 $N$。我们给出了位置引理的几个进一步应用，超出了上述内部可适应花圈产品的特征。