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

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

2011-02-15

arxiv.org/abs/1102.3138v2

Mathematics

Group Theory

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

Scientific paper

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.

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