[Paper Review] Statistical hyperbolicity in Teichmüller space

Spencer Dowdall, Moon Duchin+1 more|arXiv (Cornell University)|Aug 27, 2011

Mathematical Dynamics and Fractals9 references4 citations

TL;DR

This paper establishes that Teichmüller space exhibits statistical hyperbolicity: although not hyperbolic in the strict sense, geodesics typically spend a positive proportion of time in the thick part, and under various natural measures (Hausdorff, Busemann, Holmes–Thompson, visual, and Masur–Veech), the average distance between points on large spheres is asymptotically $2r$, indicating maximal spreading. This implies that 'most triangles are mostly thin' and that geodesics satisfy a robust thin triangle property when sufficiently thick for a positive fraction of their length.

ABSTRACT

In this paper we explore the idea that Teichmüller space is hyperbolic "on average." Our approach focuses on studying the geometry of geodesics which spend a definite proportion of time in some thick part of Teichmüller space. We consider several different measures on Teichmüller space and find that this behavior for geodesics is indeed typical. With respect to each of these measures, we show that the average distance between points in a ball of radius r is asymptotic to 2r, which is as large as possible. Our techniques also lead to a statement quantifying the expected thinness of random triangles in Teichmüller space, showing that "most triangles are mostly thin."

Motivation & Objective

• To investigate whether Teichmüller space exhibits hyperbolic-like behavior 'on average', despite not being $δ$-hyperbolic.
• To determine whether geodesics that spend a definite proportion of time in the thick part of Teichmüller space are typical under various natural measures.
• To quantify the generic thinness of triangles in Teichmüller space using a statistical spread index $E(X)$.
• To establish that the average distance between points on large spheres in Teichmüller space is asymptotically $2r$, indicating maximal spreading and statistical hyperbolicity.
• To show that the thin triangle property holds for geodesic segments that are thick for a positive proportion of their length, generalizing prior results.

Proposed method

• Define a statistical spread index $E(X) = \lim_{r\to\infty} \frac{1}{r} \int_{\mathcal{S}_r(x)\times\mathcal{S}_r(x)} d(y,z)\, d\mu_r(y)d\mu_r(z)$ to measure how fast the space spreads out.
• Use multiple measures on Teichmüller space: Hausdorff, Busemann, Holmes–Thompson, visual measures from quadratic differentials, and Masur–Veech measure.
• Show that these measures are mutually absolutely continuous and related by explicit inequalities, enabling transfer of probabilistic results across measures.
• Analyze random geodesics via random walks on the mapping class group, using the structure of the thick and thin parts of Teichmüller space.
• Apply large deviation estimates and exponential tail bounds to control the proportion of time geodesics spend outside thick sets, using functions $u_\tau$ and $s_{2i}$ to track sojourns.
• Use moment generating functions and exponential martingale techniques to bound the probability that a random walk spends more than a proportion $\rho$ of time outside a cocompact set, leading to exponential decay in $n$.

Experimental results

Research questions

• RQ1 Is it typical for geodesics in Teichmüller space to spend a positive proportion of their length in the thick part under natural measures?
• RQ2 Does the average distance between points on large spheres in Teichmüller space approach $2r$, indicating maximal spreading and statistical hyperbolicity?
• RQ3 Can the thin triangle property be extended to geodesic segments that are thick for a positive fraction of their length, not just fully thick segments?
• RQ4 Are the various natural measures on Teichmüller space—Hausdorff, Busemann, visual, Masur–Veech—mutually absolutely continuous, enabling consistent probabilistic analysis?
• RQ5 What is the expected proportion of time a random geodesic spends in the thick part, and how does this relate to the generic thinness of triangles?

Key findings

• The average distance between points on a large sphere of radius $r$ in Teichmüller space is asymptotically $2r$ under all considered measures, confirming statistical hyperbolicity.
• For any $\epsilon>0$ and $0<\theta\leq1$, there exist constants $C,L$ such that if a geodesic segment $I\subset[x,y]$ of length $\geq L$ spends at least proportion $\theta$ of its length in the $\epsilon$-thick part, then $I$ intersects the $C$-neighborhood of $[x,z]\cup[y,z]$ for all $z$, generalizing Rafi's thin triangle result.
• With respect to all studied measures—Hausdorff, Busemann, Holmes–Thompson, visual, and Masur–Veech—the probability that a random geodesic spends more than a proportion $\rho$ of its time outside a thick set decays exponentially in the length of the geodesic.
• The statistical spread index $E(X)$ equals $2$ for Teichmüller space under these measures, indicating that it is 'statistically hyperbolic' in the sense of maximal spreading.
• Random triangles in Teichmüller space are 'mostly thin' in the sense that each side lies uniformly close to the union of the other two sides, provided each side spends more than half its length in the thick part.
• The measures on Teichmüller space are mutually absolutely continuous, and their densities are uniformly bounded above and below by explicit constants, enabling robust probabilistic transfer across different geometric and measure-theoretic frameworks.