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
generalizing the invariants
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"
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
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
is polyhedral and from knowledge of
one can read
off whether or not G admits a finite presentation; indeed, conjecturally whether or
not G is of type FPm. The set
is also polyhedral when G is non-abelian and
is the fundamental group of a compact 3-manifold M
. In fact, in that case it has
a description in terms of the unit sphere of the Thurston norm on
In our generalization (sticking for now to n — 1)
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)
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
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
is all of dM if and only if the kernel of the action has type Fn.