[Paper Review] Distance in the curve graph

Ursula Hamenstaedt|arXiv (Cornell University)|Apr 20, 2011

Geometric and Algebraic Topology6 references4 citations

TL;DR

This paper establishes a method to estimate the distance between two curves in the curve graph of a finite-type surface up to a fixed multiplicative constant using Teichmüller geodesics. By detecting when curves remain at distance at least three along the geodesic, the authors construct a coarsely quasi-geodesic path in the curve graph, proving that the number of such intervals provides a uniform quasi-isometric estimate of the actual distance.

ABSTRACT

We estimate the distance in the curve graph of a surface S of finite type using Teichmueller geodesics and assuming to be able to detect curves of distance at least three.

Motivation & Objective

• To provide a uniform estimate of the distance in the curve graph up to a fixed multiplicative constant using Teichmüller geodesics.
• To overcome the lack of known bounds for the constant in coarse quasi-geodesic estimates of the curve graph via Teichmüller geodesics.
• To develop a method that relies only on detecting pairs of curves at distance at least three along a Teichmüller geodesic, avoiding explicit computation of tight geodesics.
• To establish a uniform quasi-isometric embedding of the Teichmüller geodesic's image in the curve graph via a recursive construction based on distance thresholds.
• To prove that the number of intervals where curves remain at distance ≥3 along the geodesic yields a lower bound on the curve graph distance, up to a multiplicative constant.

Proposed method

• Define a map Υ from the unit cotangent bundle of Teichmüller space to the curve graph, assigning to each quadratic differential a δ-wide curve.
• Use the fact that Υ(q_t) forms a coarsely Lipschitz unparametrized quasi-geodesic in the curve graph with uniform constants.
• Construct a sequence of times t_0 < ... < t_n along a Teichmüller geodesic such that d_CG(Υ(q_s), Υ(q_t)) ≥ 3 for s ≤ t_i and t ≥ t_{i+1}.
• Apply the triangle inequality and coarse quasi-geodesic properties to bound the total curve graph distance in terms of the number of such intervals.
• Use induction and subsurface projection arguments to control the diameter of projections and ensure that sufficiently many distinct curves are detected along the geodesic.
• Leverage results from Masur and Minsky (MM00, MM99) and Rasmussen (R10) on subsurface projections and marking complexity to derive a recursive measure of time intervals with sufficient separation.

Experimental results

Research questions

• RQ1 Can the distance in the curve graph be estimated up to a fixed multiplicative constant using only the detection of curve pairs at distance at least three along a Teichmüller geodesic?
• RQ2 What is the relationship between the number of intervals along a Teichmüller geodesic where curves remain at distance ≥3 and the actual curve graph distance?
• RQ3 How can one construct a uniform quasi-geodesic in the curve graph using the image of a Teichmüller geodesic under the Υ map?
• RQ4 To what extent does the behavior of the Υ map along Teichmüller geodesics reflect the hyperbolic geometry of the curve graph?
• RQ5 Can the lack of explicit bounds for the quasi-geodesic constant be overcome by focusing on coarse separation conditions (e.g., distance ≥3) rather than parametrized distance?

Key findings

• There exists a constant θ > 0 such that if a Teichmüller geodesic has n intervals where curves remain at distance at least three, then the curve graph distance between the endpoints is at least n/θ − θ.
• The number of such intervals provides a lower bound on the curve graph distance, with the bound uniform across all surfaces and geodesics.
• The method avoids explicit computation of tight geodesics by relying only on detecting when curves are at distance at least three, which is decidable in polynomial time.
• The construction ensures that the image of the Teichmüller geodesic under Υ yields a uniform quasi-geodesic in the curve graph, even when the parametrization is not preserved.
• The proof uses subsurface projection arguments and induction on complexity, showing that if projections into subsurfaces have large diameter, then the curve graph distance is large.
• The final estimate is achieved by showing that sufficiently many distinct curves are detected along the geodesic, ensuring that the distance grows linearly with the number of intervals.