Boundaries for CAT(0) groups

• Main Author: Iniotakis, Jan-Mark
• Published: University of Warwick 2003
• Subjects: 512.2; QA Mathematics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.399372

In this thesis we construct a boundary δG for an arbitrary CAT(0) group G. This boundary is compact and invariant under group isomorphisms. It carries a canonical (possibly trivial) G-action by homeomorphisms. For each geometric action of G on a CAT(0) space X there exists a canonical G-equivariant continuous map T : δG → δX. If G is a word-hyperbolic CAT(0) group, its boundary δG coincides with the usual Gromov boundary. If G is free abelian of rank k, its boundary is homeomorphic to the sphere Sk-1. For product groups of the types G X Zk and G x H, where G and H are non-elementary word-hyperbolic CAT(0) groups, the boundary is worked out explicitly. Finally, we prove that the marked length spectrum associated to a geometric action of a torsion-free word-hyperbolic group on a CAT(0) space determines the isometry type of the CAT(0) space up to an additive constant.