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

We exhibit 3-generator Artin groups which have finite 2-dimensional Eilenberg-Mac Lane spaces, but which do not act properly discontinuously by semi-simple isometries on a 2-dimensional CAT(0) complex. We prove that infinitely many of these groups are the fundamental groups of compact, non-positively curved 3-complexes. These examples show that the geometric dimension of a CAT(0) group may be strictly less than its CAT(0) dimension. One way of associating a dimension to a discrete group G is to take the minimal dimension of a contractible free G-CW complex or, equivalently, to take the minimal dimension of a K(G, 1) complex. This is called the geometric dimension of G. For example, groups with torsion are infinite dimensional, and the dimension of Zn is equal to n. Contractible free G-CW complexes arise naturally in geometric group theory, where the contractibility of the G-CW complex is often a consequence of non-positive curvature. Specific instances include the case when G acts freely and properly discontinuously by isometries on a CAT(0) complex, and the case when a torsion free word-hyperbolic G acts on its Rips complex. Restricting attention now to the class of CAT(0) metric spaces which have the underlying topology of a CW-complex, we define the CAT(0) dimension of a group G to be the minimal dimension of such a space on which G acts properly discontinuously by semi-simple isometries, or ∞ if no such action exists. (See section 1 for the definitions of “CAT(0)” and “semi-simple”). Clearly the ∗The first author acknowledges support from the NSF and EPSRC. The second author acknowledges the support of a UK EPSRC Research Assistantship, and a grant from the Conseil Régional de Bourgogne. Both authors wish to thank the Faculty of Mathematical Studies, University of Southampton, for support, conversations and hospitality during the preparation of this work.