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

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.