A ‘transversal’ for minimal invariant sets in the boundary of a CAT(0) group
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- by Dan P. Guralnik and Eric L. Swenson PDF
- Trans. Amer. Math. Soc. 365 (2013), 3069-3095 Request permission
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 \[ \operatorname {diam}\partial _{_\mathrm {T}} X\leq 2\pi -\arccos \left (-\frac {1}{d+1}\right )\] 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.References
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Additional Information
- Dan P. Guralnik
- Affiliation: Electric & Systems Engineering, University of Pennsylvania, 200 South 33rd Street, Philadelphia, Pennsylvania 19104
- Email: guraldan@seas.upenn.edu
- Eric L. Swenson
- Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
- Email: eric@mathematics.byu.edu
- 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.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 3069-3095
- MSC (2010): Primary 20F67, 37B05
- DOI: https://doi.org/10.1090/S0002-9947-2012-05714-X
- MathSciNet review: 3034459