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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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.
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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