CAT(\(0\)) and CAT(\(-1\)) dimensions of torsion free hyperbolic groups.
A very important question in the study of hyperbolic groups is the relationship between Gromov's \(\delta\)-hyperbolicity and the notions of CAT(\(\kappa\)). For example, it is known that a group acting properly discontinuously and cocompactly by isometries on a CAT(\(0\)) space without isometrically embedded flat planes is \(\delta\)-hyperbolic, but it is unknown whether all \(\delta\)-hyperbolic groups admit such an action on a CAT(\(0\)) space. One of the goals of the paper under review is to demonstrate that when searching for a proper CAT(\(0\)) or CAT(\(-1\)) space on which a given hyperbolic group is to act properly discontinuously and cocompactly by isometries, it may be the case that the dimension of the space is greater than the geometric dimension of the group. The authors give an example of a free-by-cyclic group with CAT(\(0\)) dimension equal to 2, but CAT(\(-1\)) dimension equal to 3. They also find an infinite family of torsion-free hyperbolic groups of geometric dimension 2 that do not act properly discontinuously by isometries on a proper CAT(\(0\)) metric space of dimension 2. The family includes (infinitely many) free-by-cyclic groups. This result means that when approaching the question of whether all \(\delta\)-hyperbolic groups are CAT(\(0\)) one has to look for CAT(\(0\)) structures above the geometric dimension, even within the class of hyperbolic free-by-cyclic groups. Although it was known that finding such a CAT(\(0\)) structure was easier when looking above the geometric dimension of the group, this theorem implies that it is sometimes necessary to do so. The paper uses a particular hyperbolic group and corresponding \(2\)-complexes in a way analogous to \(\mathbb{Z}^2\) and flat \(2\)-tori. The paper provides many of the necessary basic definitions as well as useful and beautiful illustrations.
scientific article; zbMATH DE number 5122267