A qualitative description of the horoboundary of the Teichmüller metric

Azemar, Aitor¹

¹ School of Mathematics and Statistics, University of Glasgow, Glasgow, United Kingdom

Algebraic and Geometric Topology, Tome 24 (2024) no. 7, pp. 3919-3984

Résumé

Two commonly studied compactifications of Teichmüller spaces of finite type surfaces with respect to the Teichmüller metric are the horofunction and visual compactifications. We show that these two compactifications are related, by proving that the horofunction compactification is finer than the visual compactification. This allows us to use the straightforwardness of the visual compactification to obtain topological properties of the horofunction compactification. Among other things, we show that Busemann points of the Teichmüller metric are not dense within the horoboundary, answering a question of Liu and Su. We also show that the horoboundary of Teichmüller space is path connected, determine for which surfaces the horofunction compactification is isomorphic to the visual one and show that some horocycles diverge in the visual compactification based at some point. As an ingredient in one of the proofs we show that extremal length is not C2 along some paths that are smooth with respect to the piecewise linear structure on measured foliations.

Détail

DOI : 10.2140/agt.2024.24.3919

Keywords: visual compactification, Gardiner–Masur compactification, Busemann points