Geodesically Tracking Quasi-Geodesic Paths for Coxeter Groups and CAT(0) Boundaries
by
Michael Mihalik
Vanderbilt University
Coauthors: Steven Tschantz and Kim Ruane

Let (W, S) be a finitely generated Coxeter system and L(W, S) the Cayley graph of (W, S). For general Coxeter groups, not all quasi-geodesic rays in L are tracked by geodesics. In this paper we classify the L-quasi-geodesic rays that are tracked by geodesics. As a corollary we show that if W acts geometrically on a CAT(0) space X, then CAT(0) geodesics in X are tracked by Cayley graph geodesics (where the Cayley graph is equivariantly placed in X). This last result is an important step in transforming the powerful combinatorics of Coxeter groups to results about their CAT(0) boundaries. Another corollary of our main result is that if A is a subset of S, and W acts geometrically on the CAT(0) space X, then the subgroup generated by A is quasi-convex in X.

A reflection in W is a conjugate of an element of S. The set of edges in L fixed by a reflection is a wall of L. The walls of L partition the edges of L into disjoint sets and the closure of the compliment of a wall in L has exactly two components (which are interchanged by the reflection). For a path b in L and vertex t of b let the bracket number of t in b be the number of walls Q such that there is an edge of Q on either side of t in b. Our main theorem is:

Suppose (W, S) is a finitely generated Coxeter system, and L(W, S) the Cayley graph of W with respect to S. An infinite or bi-infinite (l, e)-quasi-geodesic edge path a is tracked by an edge path geodesic iff there is a bound B on the bracket number of all vertices of a.

Date received: December 6, 2006