Geodesically tracking quasi-geodesic paths for Coxeter groups

摘要

If Λ is the Cayley graph of a Gromov hyperbolic group, then it is a fundamental fact that quasi-geodesics in Λ are tracked by geodesics. Let (W, S) be a finitely generated Coxeter system and Λ be the Cayley graph of (W, S). For general Coxeter groups, not all quasi-geodesic rays in Λ are tracked by geodesics. In this paper, we classify the Λ-quasi-geodesic rays that are tracked by geodesics. As corollaries 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 (taking the Cayley graph as equivariantly placed in X) and for any A ? S, the special subgroup is quasi-convex in X. We also show that if g is an element of infinite order for (W, S), then the subgroup is tracked by a Cayley geodesic in Λ (in analogy with the corresponding result for word hyperbolic groups).