To every Gromov hyperbolic space X one can associate a at infinity called the boundary of X. showed that quasi-isometries metric spaces induce homeomorphisms on their boundaries, thus giving rise to well-defined notion group. Croke and Kleiner visual non-positively curved (CAT(0)) groups is not well-defined, since quasi-isometric CAT(0) have non-homeomorphic boundaries. We attempt construct an ...

In this short note, we show that a group acting geometrically on CAT(0) cube complex with virtually abelian hyperplane-stabilisers must decompose as free product of groups and surface groups.

We additionally obtain a complete characterization of inner amenability for permutational wreath product groups. One the main ingredients used this is general lemma which we call location lemma, allows us to “locate” conjugation invariant means on group $G$ relative given normal subgroup $N$ $G$. give several further applications beyond aforementioned amenable products.