×

zbMATH — the first resource for mathematics

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

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
PDF BibTeX XML Cite
Full Text: DOI