Genericity of sublinearly Morse directions in general metric spaces

TL;DR

This paper demonstrates that in certain metric spaces with specific group actions, the sublinearly Morse boundary encompasses almost all points in the horofunction boundary, highlighting its genericity.

Contribution

It establishes that for proper statistically convex-cocompact actions, the sublinearly Morse boundary has full measure in the horofunction boundary, revealing its prevalence.

Findings

Sublinearly Morse boundary has full Patterson-Sullivan measure.

The result applies to proper geodesic metric spaces with specific group actions.

Highlights the genericity of sublinearly Morse directions in these spaces.

Abstract

In this paper, we show that for a proper statistically convex-cocompact action on a proper geodesic metric space, the sublinearly Morse boundary has full Patterson-Sullivan measure in the horofunction boundary.