CHAPTER 1

Introduction

A thorough outline of this memoir is given in Chapter 2 for Part I and in

Chapter 10 for Part II. In this introduction we give a quick indication of what it is

about.

1.1. Cocompact is an open condition

Let (M, d) be a simply connected proper metric space which is "non-positively

curved" (i.e., CAT(O)). For example, M might be a complete simply connected

Riemannian manifold of non-positive sectional curvature, or M might be a locally

finite affine building. Let G b e a group. We study actions of G on M by isometries.

The space of all such actions, Hom(G, Isom(M)), carries the compact-open topology

(with respect to the discrete topology on G and the compact-open topology on

Isom(M)). We emphasize that while G is discrete we are considering all actions by

isometries, not just discrete actions.

An action p : G — • Isom(M) is cocompact if there is a compact subset K of M

such that the G-translates of K cover M (i.e., GK = M). One of our results is

• Cocompactness is an open condition on p. In other words, the set of all

cocompact actions p is open in Hom(G, Isom(M)).

1.2. Controlled connectivity

We do not prove this openness theorem directly. Rather it falls out naturally as

a special case of more general considerations: it turns out that cocompactness can

be viewed as the ( —l)-dimensional case of an n-dimensional connectivity property

of p which we call "controlled n-connected", abbreviated to "CCn". Very roughly,

the flavor of this is the following: Consider a free contractible G-CW-complex

X and a G-map h : X — » M; then p is

CCn

if the /i-preimages of balls in M

are, in some sense, coarsely n-connected subsets of X. This property is extracted

and generalized from work on the "geometric invariants"

En(G)

of the group G

introduced in [BS 80], [BNS 87] [BRe 88] [Re 88], [Re 89]: these apply in the

special case where M — E m and p is an action by translations. The fact that

cocompactness is an open condition is now the special case n — 0 of the following

result:

• If G has

type1

F

n

,

CCn~l

is an open condition. See Theorem B.

There is a second special case where

CCn~1

has a familiar interpretation.

G has type Fn if there is a K(G, l)-complex with finite n-skeleton. All groups have type

Fo, Fi is "finitely generated", F2 is "finitely presented", etc.

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