Two-dimensional Artin groups with CAT(0) dimension three. (English) Zbl 1070.20043

Brady, Noel; Crisp, John

Geom. Dedicata 94, 185-214 (2002).

M. R. Bridson showed [in Math. Res. Lett. 8, No. 4, 557-567 (2001; Zbl 0990.20026)] that there exist groups with geometric dimension 2, but CAT(0) dimension 3. In the paper under review more examples of this nature are given by exhibiting 3-generator Artin groups which have finite two-dimensional Eilenberg-Mac Lane spaces, but which do not act properly discontinuously by semi-simple isometries on a two-dimensional CAT(0) complex. To prove that all but finitely many of the 3-generator Artin groups are the fundamental groups of compact nonpositively curved three-dimensional piecewise Euclidean complexes, first dihedral type Artin groups are introduced. For these, compact, non-positively curved 3-complexes are introduced, which are used as building blocks for the complexes necessary to obtain the results. Several related open questions are formulated in the end.

Reviewer: Herman J. Servatius (Worcester)

MSC:

20F36	Braid groups; Artin groups
20F67	Hyperbolic groups and nonpositively curved groups
57M07	Topological methods in group theory
57M60	Group actions on manifolds and cell complexes in low dimensions

Keywords:

Artin groups; non-positive curvature; group actions; cohomological dimension; topological dimension; discrete groups; length functions; two-dimensional complexes