Technion Geometry and Topology Seminar

NOTE SPECIAL DAY AND TIME

DATE AND TIME: Wednesday, January 12, 2011, 13:30-14:30

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.

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Technion Math. Net (TECHMATH)
Editor: Michael Cwikel   <techm@math.technion.ac.il>
Announcement from: Micha Sageev   <sageevm@math.technion.ac.il>
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