Sublinearly Morse Geodesics in CAT(0) Spaces: Lower Divergence and Hyperplane Characterization

TL;DR

This paper introduces k-lower divergence in CAT(0) spaces, providing new characterizations of k-contracting geodesic rays through divergence, slim triangles, and hyperplane separation, advancing understanding of geometric properties.

Contribution

It defines k-lower divergence and offers novel characterizations of k-contracting geodesics in CAT(0) spaces and cube complexes, linking divergence with hyperplane structure.

Findings

k-lower divergence characterizes geodesic rays

k-contracting geodesics are characterized by k-slim triangles

Hyperplane separation sequences characterize k-contracting rays

Abstract

We introduce the notion of k-lower divergence for geodesic rays in CAT(0) spaces. Building on the work of Charney and Sultan we give various characterizations of k-contracting geodesic rays using k-lower divergence and k-slim triangles. We also characterize k-contracting geodesic rays in CAT(0) cube complexes using sequences of well-separated hyperplanes.