The K(π, 1)-Problem for Hyperplane Complements Associated to Infinite Reflection Groups

Relative hyperbolicity and Artin groups

“…Moreover, the Deligne complex is 2-dimensional and is CAT(0) when equipped with the Moussong metric. These statements were all established in the paper of the first author and M. Davis [7].…”

“…More generally, any subgroup which is conjugate to a standard parabolic subgroup (of G(∆) or W (∆)) shall be referred to as a parabolic subgroup. Probably the most important tool currently used in the study of infinite type Artin groups is the Deligne complex (see [7], etc..). We described this complex in detail.…”

“…In particular, the Deligne complex is not even a locally compact space (while the Davis complex clearly is). Nevertheless, a lot of important information is carried by the action of the Artin group on its Deligne complex, as can be seen from [7,8] etc.…”

“…[5]). There are two very natural choices for such a metric (namely the Moussong metric and the cubical metric) described in [7]. These specific metrics are particularly useful when they can be shown to be nonpositively curved, or CAT(0), as demonstrated in [7], where we refer the reader for further details.…”

“…There are two very natural choices for such a metric (namely the Moussong metric and the cubical metric) described in [7]. These specific metrics are particularly useful when they can be shown to be nonpositively curved, or CAT(0), as demonstrated in [7], where we refer the reader for further details. However, in what follows, the actual choice of piecewise Euclidean or hyperbolic metric is more or less irrelevant since for any such metric d, the Deligne complex (D, d) will be (G-equivariantly) quasi-isometric to the 1-skeleton of D equipped with the unit-length edge metric.…”