3 N ov 1 99 8 Connectivity properties of group actions on non - positively curved spaces II : The geometric invariants

This Part II continues our paper [BGI], but we have made it as independent of that paper as possible. The reader familiar with the self-contained essay on finitary sheaves and finitary maps which is §4 of [BGI] can read this second paper with occasional references back to [BGI] for some details and analogies. A full outline of the present paper is found in §10 below. Let G be a group, let M be a simply connected “non-positively curved”, i.e. CAT(0), metric space, and let ρ : G → Isom(G) be an action ofG onM by isometries. In our previous paper [BGI] we introduced, when G is of type Fn, a property of the action ρ which we called “controlled (n − 1)-connectedness”, abbreviated CC. This property was defined in terms of the filtration of M by the balls Br(a) centered at some base point a ∈ M together with a free contractible G-CW-complex over M . In the present paper we pay attention to the points at infinity, i.e. the “boundary” ∂M of the spaceM . For each point e ∈ ∂M we introduce a new property of the action ρ, analogous to CC but defined using the filtration of M by horoballs centered at e rather than by the balls Br(a); we call this “CC n−1 over e”. Whereas in the previous situation the property CC was independent of the base point a ∈ M it will now depend in a delicate way upon the point e ∈ ∂M . Therefore the subset of ∂M