The local-to-global property for Morse quasi-geodesics

TL;DR

This paper establishes a local-to-global property for Morse quasi-geodesics across various groups, enabling generalizations of known theorems and revealing new geometric and algebraic insights.

Contribution

It introduces a unifying local-to-global framework for Morse quasi-geodesics applicable to multiple group classes, extending existing combination theorems and developing new tools for hyperbolic spaces.

Findings

Generalizes combination theorems for stable subgroups

Provides a Cartan-Hadamard type theorem for hyperbolicity detection

Shows discreteness of translation lengths for Morse elements

Abstract

We show the mapping class group, CAT(0) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan-Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.