Preface

This memoir presents a topological theory of (not necessarily discrete) actions

of a group G by isometries on a proper CAT(O) metric space M. Examples of such

spaces are numerous and well-known, perhaps the best known being simply con-

nected manifolds of non-positive sectional curvature. Other examples include finite

products of locally finite trees and, more generally, locally finite affine buildings.

For each such isometric action p of G on M and each n 0 we define a subset

En(p)

generalizing the invariants

En(G)

of Bieri-Neumann-Strebel-Renz (abbrevi-

ated here to BNSR) [BS 80], [BS 81], [BNS 87], [BRe 88], [Re 88], [Re 89].

The case n = 0 is only relevant for non-cocompact actions and can be ignored in

this preface. The case n = 1 can be sketched quickly and so we will concentrate on

it first, noting that everything in the memoir is done for general n.

Here is a description of the "classical"

D1(G),

often called the Bieri-Neumann-

Strebel Invariant of G. Assuming G finitely generated, let h : T — V be the

canonical G-map from the Cayley graph (with respect to a chosen finite set of

generators) to the G-vector space V := G/G' ®z H& - Choose an inner product for

V to make it a Euclidean space. Then each e G S(V), the sphere at infinity of V,

defines half spaces in V: we say that e G S1(G) if and only if the /i-preimages of

all these half spaces are connected subsets of I\ Thus

EX(G)

is a subset of S(V).

The Invariant E^G) contains the information needed to decide whether a given

subgroup S of G containing the commutator subgroup G' is finitely generated. If

G is metabelian

D1(G)

is polyhedral and from knowledge of

S1(G)

one can read

off whether or not G admits a finite presentation; indeed, conjecturally whether or

not G is of type FPm. The set

E1(G)

is also polyhedral when G is non-abelian and

is the fundamental group of a compact 3-manifold M

3

. In fact, in that case it has

a description in terms of the unit sphere of the Thurston norm on

Hi(Ms).

In our generalization (sticking for now to n — 1)

E1(p)

is described similarly:

one has a given action p : G — • Isom(M) of G on M and one chooses a G-map

h : T — M. The proper CAT(O) space M has a compactifying boundary dM and

E1(p) is defined to be a subset of dM as follows: e G E1(p) if and only if the

/z-preimages of horoballs in M centered at e are connected. Note that half-spaces

in V, above, are horoballs, so this is a generalization. When the subgroup p(G) is

discrete (as is the case in the previous action on V)

E1(p)

is all of dM if and only

if ker(p) is finitely generated.

We say that p is "controlled O-connected" over e if and only if e G E1(p).

When n replaces 1, the relevant notion is called "controlled (n — l)-connected",

i.e. preimages of horoballs at e are (n — reconnected in an approximate sense, and

we define

En(p)

to be the set of those e G dM over which the action is controlled

(n — reconnected. If p(G) is discrete in Isom(M) our Theorem A' says that

Dn(p)

is all of dM if and only if the kernel of the action has type Fn.

xi