Let G have type Fn and let the action p have discrete orbits. Then p is
if and only if the stabilizers of points of M have type Fn. See
Theorem A.
When we come down to specific examples we find new information about some
very old group actions. If m stands for a natural number and s(m) for the number
of different prime divisors of m we find
The natural action of the group G = SX2(Z[^]) on the hyperbolic plane
by Mobius transformations is
but not
The lowest case of this, s(l) = 0, is the well-known fact that SL^i^) does not act
1.3. The Boundary Criterion
The art of deciding whether a given group action p : G Isom(M) is
is still in its infancy. An approach via the compactifying boundary dM of the
CAT(0)-space M is presented in Part II. There we define the analogous concept of
p being "controlled n-connected (or
over an endpoint e G 9M". In rough
terms this means (referring back to h : X —- M) that the preimages of the horoballs
of M at e are coarsely n-connected subsets of X. The precise relationship between
the various CCn-properties is exhibited in the following result
Let M be almost geodesically complete and let G be of type Fn. Then
an isometric action p of G on M is C C n _ 1 if and only if p is CC71'1 over
each boundary point e G dM.
This "Boundary Criterion" should be viewed in the tradition of local-global
principles: It breaks up the problem of deciding whether p is CCn~l into "local"
questions over each e G dM, so that different methods and viewpoints may be
used for different kinds of endpoints. In our paper [BG] we apply it to establish
the above *SX2-theorem; we do not know of a direct proof avoiding the Boundary
Criterion. In fact, we strengthen that theorem to (see §10.7B):
The canonical SX2(Z[—])-action on
is CC°° over each irrational bound-
ary point e G dM2 and is CC3^'2 but not CC8^'1 over each rational
boundary point e.
1.4. The Geometric Invariants
Whether a given isometric action p is CCn~l over e G dM depends, in a
delicate way, upon the endpoint e. Therefore the subset of 9M,
= {e\p is
over e}
becomes interesting in its own right and in Part II we investigate its structure. A
key tool (among other things to establish the Boundary Criterion) is the subset
We prove:
(p) is an open subset of dM with respect to the Tits-metric topology
on dM, but is not, in general, open with respect to the usual (compact)
topology on dM.
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