Morse subsets of injective spaces are strongly contracting

TL;DR

This paper establishes a characterization of Morse quasi-geodesics in injective metric spaces as strongly contracting, with implications for groups acting on such spaces, including hierarchically hyperbolic groups.

Contribution

It proves that Morse and strongly contracting quasi-geodesics coincide in injective spaces and explores consequences for group actions, including acylindrical hyperbolicity.

Findings

Morse quasi-geodesics are equivalent to strongly contracting in injective spaces

Injective spaces have the Morse local-to-global property

Groups acting properly on injective spaces are acylindrically hyperbolic if they contain a Morse ray

Abstract

We show that a quasi-geodesic in an injective metric space is Morse if and only if it is strongly contracting. Since mapping class groups and, more generally, hierarchically hyperbolic groups act properly and coboundedly on injective metric spaces, we deduce various consequences relating, for example, to growth tightness and genericity of pseudo-Anosovs/Morse elements. Moreover, we show that injective metric spaces have the Morse local-to-global property and that a non-virtually-cyclic group acting properly and coboundedly on an injective metric space is acylindrically hyperbolic if and only it contains a Morse ray.