Technion Geometry and Topology Seminar NOTE SPECIAL DAY AND TIME DATE AND TIME: Wednesday, January 12, 2011, 13:30-14:30 PLACE: Amado 919 SPEAKER: Dan Guralnik (University of Oklahoma) TITLE: Dynamics of boundaries of CAT(0) groups and the rank rigidity conjecture (joint with Eric Swenson, BYU) ABSTRACT: We discuss a new approach to studying the geometry and topology of the boundary of an unbounded CAT(0) space $X$ admitting a proper co-compact action by a group $G$. There are two standing conjectures in the field, extending Ballman's rank rigidity theorem for Hadamard manifolds to general CAT(0) spaces, namely: (1) THE CLOSING LEMMA -- $G$ has rank one if and only if the Tits boundary $\partial_T X$ of $X$ has diameter $\pi$, and (2) RANK RIGIDITY -- if $X$ is geodesically-complete and irreducible (not a metric product) then either $G$ has rank one, or $X$ is a symmetric space or an affine building. Recently, (2) has been affirmed for the case when $X$ is a cubing, by Caprace and Sageev. For the general case, one direction for attacking (2) was indicated by Leeb, who showed it suffices to verify that $\partial_T X$ is a spherical building. Our project is aimed, essentially, at classifying such boundaries using the dynamics of the action of $G$ on the cone boundary $\partial X$. We show how to extend this action to an action of the Stone-\v{C}ech compactification $\beta G$ on $\partial X$ by $1$-Lipschitz operators of $\partial_T X$, actually coinciding with foldings when $X$ is a building and $G$ is a group of its automorphisms, and having `folding-like' properties in the general case. This enables us to study the interactions between the minimal closed $G$-invariant sets in $\partial X$ and the geometry of $\partial_T X$. Among other results, we prove a dynamical characterization of crystallographic groups within the class of CAT(0) groups. The aim of my talk will be to present the ingredients of our approach to conjectures (1) and (2), and explain how they are applied to produce a proof of this characterization. --------------------------------------------------------- Technion Math. Net (TECHMATH) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Micha Sageev <sageevm@math.technion.ac.il>