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

Abstract

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 ${\partial }_{\infty }X$. 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.

Accepted for publication in Trans.Amer.Math.Soc., 31 pages, 2 figures, added new dimension-dependent bound on diameter of higher rank groups

References in corpus (2)

• Rank rigidity for CAT(0) cube complexes
• A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry