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.
1
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