Now, I explain some of the more technical questions that I tangle with in my thesis. Everyone of these questions are dealt with (investigated might be a better word!) in my thesis in the case of right-angled Coxeter groups. For all these questions X is a CAT(0) space, and W is a Coxeter group and s,t are elements of W: (4-24-08)

1	Is there a geometric action of W on X?	For general groups this question is the basic question of geometric group theory in the sense that the question of which two spaces are homeomorpic is the basic question of topology. For the other questions below I assume that there exists a geodesic action of W on X.
2	Do the fixed point sets of generator separate the space?	If the space doesn't have the geodesic extension property you can always do something silly. The question is more interesting when X has the geodesic extension property.
2'	Does the fixed point set of the generators separate the space into two components?	Mike Davis shows in his book that this is always the case when W acts geometrically on a Z_2 acyclic manifold. As a consequence of this he shows that these groups are strongly rigid. As a consequence of my thesis I show sort of the opposite when I show that strongly rigid implies two components.
2''	Does the fixed point set of the generators separate the space into two convex components?	This question is largely ignored in my thesis (the convexity part), but I show that it doesn't really matter that much.
3	Does is the fixed point set of st = the intersection of the fixed point sets of s and t?	Almost the same answer as in 2). What I actually show is that there is an action with the same orbits that does. More the purposes of all these other questions that's all that matters!
4	Do the special subgroups act geometrically on some convex subset?	If 2 holds (locally ... ) then it's true. I don't know if we have "rotations" but I suspect that it's still true.
5	Is the boundary of X the same as the boundary of the Davis complex of X?	Some people in the field doubt this can hold in general even for right-angled Coxeter groups. However constructing a strict fundamental domain (see below) gives one a way to build a concrete way of going from either the Cayley graph or the Davis complex to a more arbitrary CAT(0) space
6	Can X have the geodesic extension property?	Interestingly 3) and 6) are mutually exclusive for right-angled Coxeter group with transvections (i.e. when the star of one vertex is contained in the star of another)
7	Is X a (combinatorial) building?	This is really the same as question 8' in disguise.
8	Can you point to (like saying that the fundamental domain is the region between certain fixed point sets) a fundamental domain?	This is something that I'm still working on!! Based on 8' and other things this really comes down to understanding commensibility classes of right-angled Coxeter groups.
8'	Can you point to a closed (strict) fundamental domain? In particular does one exist?	This is the MAIN part of my thesis. The answer is YES is 2 holds (locally). It allows me to understand all of these other questions.
9	If Aut(W) = Inn(W) semi direct GraphSymmetries (this is a really easy condition to check for right-angled Coxer groups) and W acts on a X with the geodesic extension property, does the action have a strict fundamntal domain?
One would hope that all actions on W on such an X would have a strict fundamemental domain, but it is isn't difficult to cook up a lot of example where this does not happen. Essentially what goes on is that the dimension of the fixed point sets is small enough where it can "hide" inside of the space.