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Busemann points are nowhere dense

Aitor Azemar, Maxime Fortier Bourque

TL;DR

This paper proves that for Teichmüller spaces with $3g+p>4$, the set of Busemann points is nowhere dense in the horoboundary of the Teichmüller metric, indicating the metric behaves far from non-positive curvature in this boundary sense. The authors construct large simplices of boundary points outside the closure of Busemann points via admissible branched covers from the five-punctured sphere and Jenkins–Strebel differentials, then use mapping class group orbits to show these points are dense in the Busemann set. The approach combines horofunction boundary analysis, extremal length and Gardiner–Masur coordinates, and strong convergence in measured foliations to control accumulation behavior. The results sharpen previous work by Miyachi and Azemar and illuminate the intricate (and non-NPC-like) topology of the Teichmüller horoboundary.

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.

Busemann points are nowhere dense

TL;DR

This paper proves that for Teichmüller spaces with , the set of Busemann points is nowhere dense in the horoboundary of the Teichmüller metric, indicating the metric behaves far from non-positive curvature in this boundary sense. The authors construct large simplices of boundary points outside the closure of Busemann points via admissible branched covers from the five-punctured sphere and Jenkins–Strebel differentials, then use mapping class group orbits to show these points are dense in the Busemann set. The approach combines horofunction boundary analysis, extremal length and Gardiner–Masur coordinates, and strong convergence in measured foliations to control accumulation behavior. The results sharpen previous work by Miyachi and Azemar and illuminate the intricate (and non-NPC-like) topology of the Teichmüller horoboundary.

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.

Paper Structure

This paper contains 10 sections, 16 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Teichmüller space and the Teichmüller metric
  4. Quadratic differentials and Teichmüller geodesics
  5. Measured foliations and extremal length
  6. The Gardiner--Masur boundary
  7. The horoboundary and the visual boundary
  8. Horofunctions outside the closure of Busemann points
  9. Simplices outside the closure
  10. Orbits of top-dimensional simplices

Key Result

Theorem 1.1

Let $g$ and $p$ be non-negative integers such that $3g+p>4$ and let $\mathcal{T}_{g,p}$ be the Teichmüller space of Riemann surfaces of genus $g$ with $p$ punctures. Then the set of Busemann points is nowhere dense in the horo-boun-dary of the Teichmüller metric on $\mathcal{T}_{g,p}$.

Figures (1)

  • Figure 1: Some double branched covers arising in the proof of \ref{['lem:preciselift']}. Marked points are indicated by crosses and unmarked branched points by dots. The arc systems used to define the double branched covers are shown in red. The numbers indicate weights on curves.

Theorems & Definitions (28)

  • Theorem 1.1
  • Proposition 3.1: azemar2021qualitative
  • Lemma 3.2: Extension of fortier2019Divergent
  • Lemma 3.3
  • proof
  • Lemma 3.4: azemar2021qualitative
  • Definition 3.5
  • Lemma 3.6
  • Lemma 4.1
  • proof
  • ...and 18 more