Finite Area Hyperbolic Surfaces Research Articles

Counting hyperbolic multigeodesics with respect to the lengths of individual components and asymptotics of Weil–Petersson volumes

Given a connected, oriented, complete, finite area hyperbolic surface $X$ of genus $g$ with $n$ punctures, Mirzakhani showed that the number of multi-geodesics on $X$ of total hyperbolic length $\leq L$ in the mapping class group orbit of a given simple or filling closed multi-curve is asymptotic as $L \to \infty$ to a polynomial in $L$ of degree $6g-6+2n$. We establish asymptotics of the same kind for countings of multi-geodesics in mapping class group orbits of simple or filling closed multi-curves that keep track of the hyperbolic lengths of individual components, proving and generalizing a conjecture of Wolpert. In the simple case we consider more precise countings that also keep track of the class of the multi-geodesics in the space of projective measured geodesic laminations. We provide a unified geometric and topological description of the leading terms of the asymptotics of all the countings considered. Our proofs combine techniques and results from several papers of Mirzakhani as well as ideas introduced by Margulis in his thesis.

Uniform bounds on harmonic Beltrami differentials and Weil–Petersson curvatures

Abstract In this article we show that for every finite area hyperbolic surface X of type {(g,n)} and any harmonic Beltrami differential μ on X, then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of μ over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil–Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil–Petersson scalar curvature over the moduli space is uniformly comparable to {-g} as the genus g goes to infinity.

SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES

Given integers$g,n\geqslant 0$satisfying$2-2g-n<0$, let${\mathcal{M}}_{g,n}$be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus$g$with$n$cusps. We study the global behavior of the Mirzakhani function$B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$which assigns to$X\in {\mathcal{M}}_{g,n}$the Thurston measure of the set of measured geodesic laminations on$X$of hyperbolic length${\leqslant}1$. We improve bounds of Mirzakhani describing the behavior of this function near the cusp of${\mathcal{M}}_{g,n}$and deduce that$B$is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of$B$to statistics of counting problems for simple closed hyperbolic geodesics.

Cutoff on hyperbolic surfaces

In this paper, we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least $$\nicefrac {1}{4}$$ , then the distances on the surface are highly concentrated around the minimal possible value of the diameter, and that the discrete random walk exhibits cutoff. This extends the results of Lubetzky and Peres (Geom Funct Anal 26(4):1190–1216, 2016. https://doi.org/10.1007/s00039-016-0382-7 ) from the setting of graphs to the setting of hyperbolic surfaces. By utilizing density theorems of exceptional eigenvalues from Sarnak and Xue (Duke Math J 64(1):207–227, 1991), we are able to show that the results apply to congruence subgroups of $$SL_{2}\left( \mathbb {Z}\right) $$ and other arithmetic lattices, without relying on the well-known conjecture of Selberg (Proc Symp Pure Math 8:1–15, 1965), thus relaxing the condition on the Laplace spectrum of a surface. Conceptually, we show the close relation between the cutoff phenomenon and temperedness of representations of algebraic groups over local fields, partly answering a question of Diaconis (Proc Natl Acad Sci 93(4):1659–1664, 1996), who asked under what general phenomena cutoff exists.

Minimal surfaces in finite volume hyperbolic 3-manifolds N and in M × S1, M a finite area hyperbolic surface

Minimal surfaces in finite volume hyperbolic 3-manifolds N and in M × S1, M a finite area hyperbolic surface

Partial sums of excursions along random geodesics and volume asymptotics for thin parts of moduli spaces of quadratic differentials

For a non-uniform lattice in SL (2,\mathbb R) , we consider excursions of a random geodesic in cusp neighborhoods of the quotient finite area hyperbolic surface or orbifold. We prove a strong law for a certain partial sum involving these excursions. This generalizes a theorem of Diamond and Vaaler for continued fractions. In the Teichmüller setting, we consider invariant measures for the SL (2,\mathbb R) action on the moduli spaces of quadratic differentials. By the work of Eskin and Mirzakhani, these measures are supported on affine invariant submanifolds of a stratum of quadratic differentials. For a Teichmüller geodesic random with respect to a SL (2,\mathbb R) -invariant measure, we study its excursions in thin parts of the associated affine invariant submanifold. Under a regularity hypothesis for the invariant measure, we prove similar strong laws for certain partial sums involving these excursions. The limits in these laws are related to the volume asymptotic of the thin parts. By Siegel–Veech theory, these are given by various Siegel–Veech constants. As a direct consequence, we show that the word metric of mapping classes that approximate a Teichmüller geodesic ray that is random with respect to the Masur–Veech measure, grows faster than T log T .

The local maxima of maximal injectivity radius among hyperbolic surfaces

The function on the Teichmueller space of complete, orientable, finite-area hyperbolic surfaces of a fixed topological type that assigns to a hyperbolic surface its maximal injectivity radius has no local maxima that are not global maxima.

Systoles and kissing numbers of finite area hyperbolic surfaces

We study the number and the length of systoles on complete finite area orientable hyperbolic surfaces. In particular, we prove upper bounds on the number of systoles that a surface can have (the so-called kissing number for hyperbolic surfaces). Our main result is a bound which only depends on the topology of the surface and which grows subquadratically in the genus.

On largeness and multiplicity of the first eigenvalue of finite area hyperbolic surfaces

We apply topological methods to study the smallest non-zero number $\lambda_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set $\{S \in {\mathcal{M}_2}: {\lambda_1}(S) > 1/4 \}$ is unbounded and disconnects the moduli space ${\mathcal{M}_2}$.

Systole growth for finite area hyperbolic surfaces

We are interested in the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature $(g,n)$. This maximum is shown to be strictly increasing in terms of the number of cusps for small values of $n$. We also show that this function is greater than a function that grows logarithmically in function of the ratio $g/n$.

Shearing coordinates and convexity of length functions on Teichmüller space

We prove that there are Fenchel-Nielsen coordinates for the Teichmüller space of a finite area hyperbolic surface with respect to which the length functions are convex.

Orthospectra of geodesic laminations and dilogarithm identities on moduli space

Given a measured lamination on a finite area hyperbolic surface we consider a natural measure Mon the real line obtained by taking the push-forward of the volume measure of the unit tangent bundle of the surface under an intersection function associated with the lamination. We show that the measure M gives summation identities for the Rogers dilogarithm function on the moduli space of a surface.

Geodesic cusp excursions and metric diophantine approximation

We derive several results that describe the rate at which a generic geodesic makes excursions into and out of a cusp on a finite area hyperbolic surface and relate them to approximation with respect to the orbit of infinity for an associated Fuchsian group. This provides proofs of some well known theorems from metric diophantine approximation in the context of Fuchsian groups. It also gives new results in the classical setting.

Weil–Petersson isometries via the pants complex

We extend a theorem of Masur and Wolf which states that given a finite area hyperbolic surface S, every isometry of the Teichmuller space for S with the Weil-Petersson metric is induced by an element of the mapping class group for S. Our argument handles the previously untreated cases of the four times-punctured sphere, the once-punctured torus, and the twice-punctured torus.

Uniform estimates for closed geodesics and homology on finite area hyperbolic surfaces

We study the distribution of closed geodesics in homology on a finite area hyperbolic surface. We obtain an estimate which is uniform as the homology class varies, refining an asymptotic formula due to C. Epstein.