Parabolic isometries of CAT(0) spaces and CAT(0) dimensions

Abstract

We study discrete groups from the view point of a dimension gap in connection to CAT(0) geometry. Developing studies by Brady-Crisp and Bridson, we show that there exist finitely presented groups of geometric dimension 2 which do not act properly on any proper CAT(0) spaces of dimension 2 by isometries, although such actions exist on CAT(0) spaces of dimension 3.Another example is the fundamental group, G, of a complete, non-compact, complex hyperbolic manifold M with finite volume, of complex-dimension n ≥ 2. The group G is acting on the universal cover of M , which is isometric to H n C . It is a CAT(−1) space of dimension 2n. The geometric dimension of G is 2n − 1. We show that G does not act on any proper CAT(0) space of dimension 2n − 1 properly by isometries.We also discuss the fundamental groups of a torus bundle over a circle, and solvable Baumslag-Solitar groups.

CAT(0) and CAT(-1) dimensions of torsion free hyperbolic groups

“…On the other hand, we do not suppose that the actions are semi-simple. This is similar to the 2-dimensional Torus Theorem of Fujiwara, Shioya and Yamagata [14] which includes the dimension restriction, but does not require semi-simplicity. In fact we use Proposition 4.4 of [14] explicitly in order to remove any co-compactness or semi-simplicity hypothesis from our arguments (see Section 4.1).…”

“…By the work of Bridson [10] and of Brady-Crisp [5] (see also [12], [14]) one knows that the minimal dimension of a CAT(0) structure for a CAT(0) group, may be strictly greater than its geometric dimension. However, all these papers used some version of the Flat Torus Theorem, and make heavy use of the presence of periodic flats in 2-dimensional CAT(0) spaces.…”

“…This is in contrast to previous works [5], [10], [12] where semi-simplicity is assumed (because it is needed to apply the usual Flat Torus Theorem). In these cases this hypothesis can be removed by using instead the 2-dimensional Torus Theorem of [14], at the expense of supposing that the action is on a proper CAT(0) space.…”

A note on nilpotent subgroups of automorphism groups of RAAGs

“…This is essentially what we have shown in the previous proof. An alternative argument can be found in [10,Corollary 5.1], where it is furthermore proved that, for any proper action of H 3 (Z) on a CAT(0) space, C is necessarily parabolic. It is worth noticing that H 3 (Z) has a proper parabolic action on the complex hyperbolic plance H 2 C , which is a proper finite-dimensional CAT(−1) space [10, Corollary 5.1].…”

Three-dimensional FC Artin Groups are CAT(0)

“…But, quite often, this is the case. If G is virtually torsion-free and acts properly, cocompactly, and cellularly on a CAT(0) cell complex X, then, since CAT(0) spaces are contractible, the dimension of X is bounded below by the virtual cohomological dimension of G. In fact, there are examples of groups (in particular, some three generator Artin groups) where the cohomological dimension is two, the group admits a geometric action on a three-dimensional CAT(0) space, but the group does not admit a geometric action on any two-dimensional CAT(0) space [6,11,26,28].…”