A qualitative description of the horoboundary of the Teichmüller metric
arXiv:2108.11698 · doi:10.2140/agt.2024.24.3919
Abstract
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 simplicity of the visual compactification to obtain topological properties of the horofunction compactification. Among other things, we show that the horoboundary of Teichmüller space is path connected and that its Busemann points are not dense, we determine for which surfaces the horofunction compactification is isomorphic to the visual one, and we 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 along some paths that are smooth with respect to the piecewise linear structure on measured foliations.
52 pages, 14 figures. Updated to match a accepted version. Improved continuity result from extremal length not being \(C^{2+\varepsilon}\) to not being \(C^{2}\) along some paths