Busemann points are nowhere dense

Abstract

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

22 pages, 1 figure