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Thurston's metric on Teichmüller space and the translation distances of mapping classes
- Source :
- Annales Academiae Scientiarum Fennicae. Mathematica, Annales Academiae Scientiarum Fennicae. Mathematica, Academia Scientiarum Fennica, 2016, 41, pp.867--879. ⟨10.5186/aasfm.2016.4158⟩
- Publication Year :
- 2016
- Publisher :
- HAL CCSD, 2016.
-
Abstract
- We show that the Teichmuller space of a surface without boundary and with punctures, equipped with the Thurston metric, is the limit in an appropriate sense of Teichmuller spaces of surfaces with boundary, equipped with their arc metrics, when the boundary lengths tend to zero. We use this to obtain a result on the translation distances of mapping classes for their actions on Teichmuller spaces equipped with the Thurston metric. In this paper, we show that the arc metrics on the Teichmuller space of surfaces with boundary limit to the Thurston metric on the Teichmuller space of a surface without boundary, by making the boundary lengths tend to zero. We use this to prove a result on the translation distances for mapping classes. We introduce some notation before stating precisely the results. In all this paper, S = Sg,p,n is a connected orientable surface of finite type, of genus g with p punctures and n boundary components. We assume that S has negative Euler characteristic, i.e., χ(S) = 2 − 2g − p − n 0, we denote by ∂S the boundary of S. A hyperbolic structure on S is a complete metric of constant curvature −1 such that (i) each puncture has a neighborhood isometric to a cusp, i.e., to the quotient {z = x + iy ∈ H 2 | y > a}/hz 7→z + 1i
- Subjects :
- Teichmüller space
Quasiconformal mapping
Pure mathematics
General Mathematics
Boundary (topology)
01 natural sciences
hyperbolic geometry
symbols.namesake
Thurston metric
Hyperbolic set
Euler characteristic
[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT]
0103 physical sciences
0101 mathematics
Mathematics
translation distance
010102 general mathematics
Mathematical analysis
quasiconformal mapping
32G15
30F60
mapping class group
Surface (topology)
Mathematics::Geometric Topology
arc metric
Mapping class group
pseudo-Anosov
Metric (mathematics)
reducible
symbols
010307 mathematical physics
Subjects
Details
- Language :
- English
- ISSN :
- 1239629X
- Database :
- OpenAIRE
- Journal :
- Annales Academiae Scientiarum Fennicae. Mathematica, Annales Academiae Scientiarum Fennicae. Mathematica, Academia Scientiarum Fennica, 2016, 41, pp.867--879. ⟨10.5186/aasfm.2016.4158⟩
- Accession number :
- edsair.doi.dedup.....cd707211f4ac6db4fc611ace66796695