Morse actions of discrete groups on symmetric spaces: Local-to-global principle

TL;DR

This paper establishes a local-to-global principle for Morse quasigeodesics and actions, enabling the recognition and construction of Morse subgroups in higher rank symmetric spaces through geometric methods.

Contribution

It introduces a new local-to-global principle for Morse actions and quasigeodesics, and develops geometric techniques for constructing Morse subgroups without Tits' ping-pong.

Findings

Algorithmic recognizability of Morse actions

Construction of Morse Schottky subgroups in higher rank groups

Purely geometric approach to Morse subgroup construction

Abstract

Our main result is a local-to-global principle for Morse quasigeodesics, maps and actions. As an application of our techniques we show algorithmic recognizability of Morse actions and construct Morse ``Schottky subgroups'' of higher rank semisimple Lie groups via arguments not based on Tits' ping-pong. Our argument is purely geometric and proceeds by constructing equivariant Morse quasiisometric embeddings of trees into higher rank symmetric spaces.