A quantified local-to-global principle for Morse quasigeodesics

TL;DR

This paper provides explicit quantitative criteria for the local-to-global principle of Morse quasigeodesics in symmetric spaces, making the verification of Anosov representations algorithmically effective and applicable to explicit examples.

Contribution

It introduces the first explicit estimates for the local-to-global principle of Morse quasigeodesics, enhancing the algorithmic verification of Anosov representations.

Findings

Derived explicit criteria for local-to-global Morse quasigeodesic behavior

Developed an effective algorithm for verifying Anosov property

Computed explicit perturbation neighborhoods for specific examples

Abstract

In arXiv:1403.7671, Kapovich, Leeb and Porti gave several new characterizations of Anosov representations $\Gamma \to G$, including one where geodesics in the word hyperbolic group $\Gamma$ map to "Morse quasigeodesics" in the associated symmetric space $G/K$. In analogy with the negative curvature setting, they prove a local-to-global principle for Morse quasigeodesics and describe an algorithm which can verify the Anosov property of a given representation in finite time. However, some parts of their proof involve non-constructive compactness and limiting arguments, so their theorem does not explicitly quantify the size of the local neighborhoods one needs to examine to guarantee global Morse behavior. In this paper, we supplement their work with estimates in the symmetric space to obtain the first explicit criteria for their local-to-global principle. This makes their algorithm for verifying the Anosov property effective. As an application, we demonstrate how to compute explicit perturbation neighborhoods of Anosov representations with two examples.