Comparison between Teichmüller and Lipschitz metrics

By Young-Eun Choi, Kasra Rafi. 2005

74 citations recorded by the source corpus.

We study the Lipschitz metric on a Teichmüller space (defined by Thurston) and compare it with the Teichmüller metric. We show that in the thin part of the Teichmüller space the Lipschitz metric is approximated up to a bounded additive distortion by the sup‐metric on a product of lower‐dimensional spaces (similar to the Teichmüller metric as shown by Minsky). In the thick part, we show that the two metrics are equal up to a bounded additive error. However, these metrics are not comparable in general; we construct a sequence of pairs of points in the Teichmüller space, with distances that approach zero in the Lipschitz metric while they approach infinity in the Teichmüller metric.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d3700b3f6e0d142a9136ce7e46b73425eb2c75e1

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On the growth of $L^2$-invariants for sequences of lattices in Lie groups (2012, background): “Since the Teichmuller distance between (Σ, γ(0) −1 ∗ d) and (Σ, d0) is bounded by C, there is a C-lipschitz map (Σ, d0)→ (Σ, γ(0) ∗ d) homotopic to the identity map [34].”
Hyperbolicity in Teichmüller space (2010, unknown): “(Examples can easily be produced using Thurston’s construction of minimal stretch maps [20] and the results in [2].”
The geometry of the disk complex (2010, background): “Corollary D of [12] gives a more precise relation between projection distance and intersection number.”
The Farrell–Jones Conjecture for mapping class groups (2016, background): “[20, 65] For every M there is I such that the following holds.”
The horoboundary and isometry group of Thurston's Lipschitz metric (2010, background): “The literature includes [16, 23, 18, 24, 19, 25, 5, 26]; most of it is very recent.”