A Combinatorial Model for the Teichmüller Metric

Abstract

We study how the length and the twisting parameter of a curve change along a Teichmüller geodesic. We then use our results to provide a formula for the Teichmüller distance between two hyperbolic metrics on a surface, in terms of the combinatorial complexity of curves of bounded lengths in these two metrics.

Lines of minima are uniformly quasigeodesic

“…(The relative twist d α (ν 1 , ν 2 ) agrees up to an additive constant with the definition of subsurface distance between the projections of |ν 1 | and |ν 2 | to the annular cover of S with core α, as defined in [Masur and Minsky 2000, Section 2.4] and used throughout [Rafi 2005;2007].) Rafi [2007] (see also [Choi et al 2006, Section 5.4]) introduced a similar notion of the twist tw q (ν, α) with respect to a quadratic differential metric q compatible with σ and proved the following result, which enters into the proof of Theorem 4.1:…”

Book Review: A primer on mapping class groups

“…They go on to estimate distances in Mod(S) in terms of distances in the curve complexes of S and its subsurfaces. A way to succinctly phrase their result was given in [11]: Mod(S) acts with quasi-isometrically embedded orbits on a finite product of δ-hyperbolic spaces; each of the hyperbolic spaces is quasi-isometric to a tree of curve complexes of subsurfaces of S. The Masur-Minsky theory led to geometric results about Mod(S): quasi-isometric rigidity [7], rapid decay property [8], measure rigidity [51], boundary amenability [29], finiteness of asymptotic dimension [11], the structure of asymptotic cones [5,6], bounds on the conjugacy problem [72], and others, and it also led to a qualitative understanding of the geometry of Teichmüller space [70].…”

How to read the length of a braid from its curve diagram

“…An essential portion of the answer seems to be provided by the insight of Masur and Minsky [11] that for any braid (or mapping class) there are only finitely many subsurfaces whose interior is tangled by the action of the braid, and by the MasurMinsky-Rafi distance formula [12]. It is, however, not clear how these results can be used in practice to prove, for instance, that the Bressaud normal form [1,3] or the transmission-relaxation normal form [4] are quasi-geodesic, or that all braids have -consistent representatives of linearly bounded length [7].…”

“…This answer, however, is false -see for instance [12] and [4]. In these papers a relation was established between a certain distorted word length and the above-mentioned intersection number, which is in turn related to the distance in Teichmüller space between a base point and its image under the braid action.…”