Busemann points are nowhere dense

TL;DR

This paper demonstrates that Busemann points form a nowhere dense set in the horoboundary of the Teichmüller metric for higher-dimensional Teichmüller spaces, indicating the metric's deviation from non-positive curvature.

Contribution

It establishes that Busemann points are nowhere dense in the horoboundary of the Teichmüller metric for complex dimensions greater than one, revealing geometric properties of the space.

Findings

Busemann points are nowhere dense in the horoboundary.

Teichmüller metric lacks non-positive curvature properties.

Results hold for all higher-dimensional Teichmüller spaces.

Abstract

We prove that the set of Busemann points (the limits of almost-geodesic rays) is nowhere dense in the horoboundary of the Teichm\"uller metric for all Teichm\"uller spaces of complex dimension strictly larger than 1. This shows that the Teichm\"uller metric is far from having non-positive curvature in a certain sense.