Quasi-Convex Subgroups of CAT(0) groups
by
Kim Ruane
Vanderbilt University

In the setting of word hyperbolic groups the definition of quasi-convex subgroup is a well-defined in the sense that once a subgroup is shown to be quasi-convex in one Cayley graph for the group, it is quasi-convex in any Cayley graph. In other words, if the group is acting geometrically on a space, then the orbit of the subgroup sits "nicely" inside the space. This also shows that a quasi-convex subgroup of a word hyperbolic group is again word hyperbolic. The same phenomenon does not happen when one moves to the setting of CAT(0) groups. With this in mind, we define a subgroup of a CAT(0) group to be quasi-convex if the orbit of the subgroup sits "nicely" inside X. In particular, we want quasi-convex subgroups to again be CAT(0) groups. This may seem like a lot to ask of a subgroup, but there are very natural examples for which the idea works. For example, the Flat Torus Theorem implies that abelian subgroups are always quasi-convex with this notion - independent of what CAT(0) space the group is acting on.

Date received: February 23, 1998