Non-Positive Curvature and Group Theory

Abstract

We have already seen that one can say a good deal about the structure of groups which act properly by isometries on CAT(0) spaces, particularly if the action is cocompact. One of the main goals of this chapter is to add further properties to the list of things that we know about such groups. In particular, in Section 1, we shall show that if a group acts properly and cocompactly by isometries on a CAT(0) space, then the group has a solvable word problem and a solvable conjugacy problem. Decision problems also form the focus of much of Section 5, the main purpose of which is to demonstrate that the class of groups which act properly by semi-simple isometries on complete CAT(0) spaces is much larger and more diverse than the class of groups which act properly and cocompactly by isometries on CAT(0) spaces; this diversity can already be seen among the finitely presented subgroups of groups that act cocompactly.