JournalscmhVol. 82, No. 1pp. 61–85

CAT(0) and CAT(-1) dimensions of torsion free hyperbolic groups

  • Noel Brady

    University of Oklahoma, Norman, United States
  • John Crisp

    Université de Bourgogne, Dijon, France
CAT(0) and CAT(-1) dimensions of torsion free hyperbolic groups cover
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Abstract

We show that a particular free-by-cyclic group G has CAT(0) dimension equal to 2, but CAT(-1) dimension equal to 3. Starting from a fixed presentation 2-complex we define a family of non-positively curved piecewise Euclidean “model” spaces for G, and show that whenever the group acts properly discontinuously by isometries on any proper 2-dimensional CAT(0) space X there exists a G-equivariant map from the universal cover of one of the model spaces to X which is locally isometric off the 0-skeleton and injective on vertex links.

From this we deduce bounds on the relative translation lengths of various elements of G acting on any such space X by first studying the geometry of the model spaces. By taking HNN-extensions of G we then produce an infinite family of 2-dimensional hyperbolic groups which do not act properly discontinuously by isometries on any proper CAT(0) metric space of dimension 2. This family includes a free-by-cyclic group with free kernel of rank 6.

Cite this article

Noel Brady, John Crisp, CAT(0) and CAT(-1) dimensions of torsion free hyperbolic groups. Comment. Math. Helv. 82 (2007), no. 1, pp. 61–85

DOI 10.4171/CMH/85