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

Dan Guralnik; Eric L. Swenson

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

Group Theory 2015-03-19 v2 Differential Geometry Dynamical Systems Metric Geometry

Authors: Dan Guralnik , Eric L. Swenson

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 whose boundary sphere intersects every minimal $G$ -invariant subset of $\partial_\infty X$ . As a result we derive a necessary and sufficient dynamical condition for $G$ to be virtually-Abelian, as well as a new approach to Ballmann's rank rigidity conjecture.

Keywords

cat(0) group boundary dynamics minimal invariant set flat virtually-abelian rank rigidity

Cite

@article{arxiv.1102.3138,
  title  = {A `transversal' for minimal invariant sets in the boundary of a CAT(0) group},
  author = {Dan Guralnik and Eric L. Swenson},
  journal= {arXiv preprint arXiv:1102.3138},
  year   = {2015}
}

Comments

Accepted for publication in Trans.Amer.Math.Soc., 31 pages, 2 figures, added new dimension-dependent bound on diameter of higher rank groups

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

Abstract

Keywords

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