A characterization of short curves of a Teichmuller geodesic

By Kasra Rafi. 2004

114 citations recorded by the source corpus.

We provide a combinatorial condition characterizing curves that are short along a Teichmuller geodesic. This condition is closely related to the condition pro- vided by Minsky for curves in a hyperbolic 3-manifold to be short. We show that short curves in a hyperbolic manifold homeomorphic to S×R are also short in the corresponding Teichmuller geodesic, and we provide examples demon- strating that the converse is not true.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0a24cbc605834dff0f887b481c062b40cab36893

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Train tracks and the geometry of the complex of curves (2005, background): “Moreover, by Theorem 5.6 of [20], there is a constant K > 0 with the following property.”
Uniform hyperbolicity of the curve graph via surgery sequences (2013, background): “In recent years the curve graph has emerged as the central object in a variety of areas, such as Kleinian groups [15, 14, 7], Teichmüller spaces [16, 17, 6] and mapping class groups [13, 2].”
Length spectra and degeneration of flat metrics (2009, unknown): “See [27] for more details.”
Shadows Of Mapping Class Groups: Capturing Convex Cocompactness (2005, background): “Rafi has characterized the short curves in hyperbolic structures on a Teichmüller geodesic in terms of the intersections of the subsurface projections of its defining laminations [59].”
Comparison between Teichmüller and Lipschitz metrics (2005, unknown): “Then we have (see, for example, [8]) qt(α) ∗ ht(α) + vt(α).”
On the growth of $L^2$-invariants for sequences of lattices in Lie groups (2012, methodology): “Combining theorems of Minsky [83] and Rafi [97], we see that as α lies in the ǫ-thick part of T(Σ) there is a lower bound inj(N) ≥ ǫ > 0 for the injectivity radius of N .”
THICK-THIN DECOMPOSITION FOR QUADRATIC DIFFERENTIALS (2014, methodology): “Following the discussion in [Raf05b], we can ignore this difficulty and treat these special geodesics as we would any other geodesic.”
Hyperbolicity in Teichmüller space (2010, background): “The following is the restatement of results in [15] in a way that is more suitable for our purposes.”
Geometry of the complex of curves and of Teichmüller space (2005, background): “Every essential closed curve c on S is freely homotopic to a closed geodesic with respect to q, and the length of such a geodesic η is the infimum of the q-lengths of any curve freely homotopic to η (compare [26] for a detailed discussion of the”