A `transversal' for minimal invariant sets in the boundary of a CAT(0) group

We introduce new techniques for studying boundary dynamics of CAT(0) groups. For a group $G$ acting geometrically on a CAT(0) space $X$ we show there is a flat $F\subset X$ of maximal dimension whose boundary sphere intersects every minimal $G$-invariant subset of ${\mathrm{\partial }}_{\mathrm{\infty }}X$. As a result we derive a necessary and sufficient dynamical condition for $G$ to be virtually-Abelian, as well as a new approach to Ballmann’s rank rigidity conjecture.