Geodesically Tracking Quasi-geodesic Paths for Coxeter Groups

TL;DR

This paper classifies quasi-geodesics in Coxeter groups that are tracked by geodesics and explores their implications for geometric actions on CAT(0) spaces, revealing new quasi-convexity properties.

Contribution

It provides a classification of tracked quasi-geodesics in Coxeter groups and establishes their geometric properties in CAT(0) spaces.

Findings

Coxeter group actions on CAT(0) spaces have CAT(0) rays tracked by geodesics.

All special subgroups of Coxeter groups are quasi-convex in the space.

Elements of infinite order in Coxeter groups are tracked by geodesics.

Abstract

The main theorem of this paper classifies the quasi-geodesics in a Coxeter group that are tracked by geodesics. As corollaries, we show that if a Coxeter group acts geometrically on a CAT(0) space X then CAT(0) rays (and lines) are tracked by Cayley graph geodesics, all special subgroups of the Coxeter group are quasi-convex in X, and in Cayley graphs for Coxeter groups, elements of infinite order are tracked by geodesics.