A Morse Lemma for quasigeodesics in symmetric spaces and euclidean buildings

TL;DR

This paper establishes a Morse Lemma for quasigeodesics in symmetric spaces and Euclidean buildings, providing a geometric characterization of Morse subgroups as undistorted, coarsely uniformly regular, and word hyperbolic.

Contribution

It introduces a Morse Lemma for quasigeodesics in nonpositively curved spaces and characterizes Morse subgroups as undistorted, coarsely uniformly regular, and hyperbolic.

Findings

Morse Lemma for quasigeodesics in symmetric spaces and Euclidean buildings

Characterization of Morse subgroups as undistorted and coarsely uniformly regular

Morse subgroups are shown to be word hyperbolic

Abstract

We prove a Morse Lemma for coarsely regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings X. The main application is a simpler coarse geometric characterization of Morse subgroups of the isometry groups Isom(X) as undistorted subgroups which are coarsely uniformly regular. We show furthermore that they must be word hyperbolic. We introduced this class of discrete subgroups in our earlier paper, in the context of symmetric spaces, where various equivalent geometric and dynamical characterizations of word hyperbolic Morse subgroups were established, including the Anosov subgroup property.