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

*(English)*Zbl 1070.20043M. 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 |