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

Authors: Dan P. Guralnik and Eric L. Swenson
Journal: Trans. Amer. Math. Soc. 365 (2013), 3069-3095
MSC (2010): Primary 20F67, 37B05
Published electronically: September 19, 2012
MathSciNet review: 3034459
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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 (denote it by $ d$), whose boundary sphere intersects every minimal $ G$-invariant subset of $ \partial _\infty X$. As applications we obtain an improved dimension-dependent bound

$\displaystyle \operatorname {diam}\partial _{_\mathrm {T}} X\leq 2\pi -\arccos \left (-\frac {1}{d+1}\right )$

on the Tits-diameter of $ \partial X$ for non-rank-one groups, a necessary and sufficient dynamical condition for $ G$ to be virtually Abelian, and we formulate a new approach to Ballmann's rank rigidity conjectures.

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Additional Information

Dan P. Guralnik
Affiliation: Electric & Systems Engineering, University of Pennsylvania, 200 South 33rd Street, Philadelphia, Pennsylvania 19104

Eric L. Swenson
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

Received by editor(s): February 14, 2011
Received by editor(s) in revised form: June 24, 2011, and September 24, 2011
Published electronically: September 19, 2012
Additional Notes: This work was partially supported by a grant from the Simons Foundation (209403 to the second author), and carried out while the first author was a post-doctoral fellow at the University of Oklahoma Mathematics Department.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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