Noncompact Hyperbolic Surface Research Articles

Shrinking target equidistribution of horocycles in cusps

Consider a shrinking neighborhood of a cusp of the unit tangent bundle of a noncompact hyperbolic surface of finite area, and let the neighborhood shrink into the cusp at a rate of T^{-1} as T rightarrow infty . We show that a closed horocycle whose length ell goes to infinity or even a segment of that horocycle becomes equidistributed on the shrinking neighborhood when normalized by the rate T^{-1} provided that T/ell rightarrow 0 and, for any delta >0, the segment remains larger than max left{ T^{-1/6},left( T/ell right) ^{1/2}right} left( T/ell right) ^{-delta }. We also have an effective result for a smaller range of rates of growth of T and ell . Finally, a number-theoretic identity involving the Euler totient function follows from our technique.

Geometric Schottky groups and non-compact hyperbolic surfaces with infinite genus

The topological type of a non-compact Riemann surface is determined by its ends space and the ends having infinite genus. In this paper for a non-compact Riemann Surface Sm,s with s ends and exactly m of them with infinite genus, such that m,s∈N and 1<m≤s, we give a precise description of the infinite set of generators of a Fuchsian (geometric Schottky) group Γm,s such that the quotient space H/Γm,s is homeomorphic to Sm,s and has infinite area. For this construction, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.

A Prime Geodesic Theorem of Gallagher Type for Riemann Surfaces

We consider a cofinite Fuchsian group of the first kind with finitely many inequivalent parabolic elements and a unitary multiplier system of an arbitrary weight on it. Through the Gallagher–Koyama approach to the prime geodesic theorem on the corresponding noncompact hyperbolic surface, we reduce the exponent in the error term from $$\frac{3}{4}$$ to $$\frac{7}{10}$$ outside a set of finite logarithmic measure. Recent advances in results of the latter type and the methods applied are briefly discussed.

Asymptotic windings of horocycles

We analyze the scaling limits of the winding process for horocycles on noncompact hyperbolic surfaces with finite area. Initial conditions with precompact forward geodesics have scaling limits with gaussian and Cauchy components. Typical initial conditions have different scaling limits along different subsequences of times, and all such scaling limits can be described. Some of our results extend to other unipotent flows.

The Teichmüller space of the Hirsch foliation

We prove that the Teichmüller space of the Hirsch foliation (a minimal foliation of a closed 3-manifold by non-compact hyperbolic surfaces) is homeomorphic to the space of closed curves in the plane. This allows us to show that the space of hyperbolic metrics on the foliation is a trivial principal fiber bundle. And that the structure group of this bundle, the arc-connected component of the identity in the group of homeomorphisms which are smooth on each leaf and vary continuously in the smooth topology in the transverse direction of the foliation, is contractible.

On the deviation of ergodic averages for horocycle flows

We give effective bounds on the deviation of ergodic averages for the horocycle flow on the unit tangent bundle of a noncompact hyperbolic surface of finite area. The bounds depend on the small eigenvalues of the Laplacian and on the rate of excursion into cusps for the geodesic corresponding to the given initial point. We also prove Ω-results which show that in a certain sense our bounds are essentially the best possible for any given initial point.

Eigenvalues of Laplacian with Constant Magnetic Field on Non-Compact Hyperbolic Surfaces with Finite Area

We consider a magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$ on a noncompact hyperbolic surface $\mM $ with finite area. $A$ is a real one-form and the magnetic field $dA$ is constant in each cusp. When the harmonic component of $A$ satifies some quantified condition, the spectrum of $-\Delta_A$ is discrete. In this case we prove that the counting function of the eigenvalues of $-\Delta_{A}$ satisfies the classical Weyl formula, even when $dA=0. $

Affine deformations of a three-holed sphere

Associated to every complete affine 3–manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface Σ. We classify these complete affine structures when Σ is homeomorphic to a three-holed sphere. In particular, for every such complete hyperbolic surface Σ, the deformation space identifies with two opposite octants in ℝ3. Furthermore every M admits a fundamental polyhedron bounded by crooked planes. Therefore M is homeomorphic to an open solid handlebody of genus two. As an explicit application of this theory, we construct proper affine deformations of an arithmetic Fuchsian group inside Sp(4,ℤ).

On the Asymptotic Behavior of Counting Functions Associated to Degenerating Hyperbolic Riemann Surfaces

In this article we will study what we call weighted counting functions on hyperbolic Riemann surfaces of finite volume. IfMis compact, then we define the weighted counting function forw⩾0 to beNM,w(T)=∑λn⩽T(T−λn)w,where {λn} is the set of eigenvalues of the Laplacian which acts on the space of smooth functions onM. IfMis non-compact, then we define the weighted counting functionNM,w(T) via the inverse Laplace transform from which one can express the weighted counting function in terms of spectral data associated toM(see Proposition 5.1 and Remark 5.2). Using the convergence results from [JL2] concerning the regularized heat trace on finite volume hyperbolic Riemann surfaces, we shall prove the following results. Theorem.Let Mldenote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface M0.LetSldenote the set of lengths of pinching geodesics on the family of surfaces Ml.For w⩾0and T⩾1/4,letGl,w(T)=2Γ(w+1)(16π)1/2∑n=1∞∑lk∈S1lksinfh(nlk/2)T−1/4(nlk/2)2(w+1/2)/2Jw+1/2(nlk(T−1/4)),where Js(x)is the J-Bessel function and Γ(x)is the standard gamma function. For0⩽T<1/4,defineGl,w(T)to be zero.Then: (a)Forw>3/2andT>0,we haveNMl,w(T)=Gl,w(T)+NM0,w(T)+o(1)asl→0;(b)For w⩾0andT>0,we haveNMl,w(T)=Gl,w(T)+o∑lk∈Sllog(1/lk)asl→0;:(c)For w⩾0andT⩾1/4,we haveGl,w(T)=2Γ(w+1)(T−1/4)w+1/2(4π)1/2Γ(w+3/2)∑lk∈Sllog(1/lk)+O(1)asl→0.Observe that by takingT<1/4, part (a) proves convergence of certain symmetric functions of the small eigenvalues, thus we prove the main result from [CC]. We discuss a method to improve the error terms in asymptotic expansion of counting functions for 0⩽w⩽3/2; see Remark 2.2 and Remark 3.6 for further discussion of the organization of this article.

Ground state and lowest eigenvalue of the Laplacian for non-compact hyperbolic surfaces

LetM be a complete Riemannian surface with constant curvature −1, infinite volume, and a finitely generated fundamental group. Denote by λ(M) the lowest eigenvalue of the Laplacian onM, and let Φ M be the associated eigenfunction. We estimate the size of λ(M) and the shape of Φ M by a finite procedure which has an electrical circuit analogue. Using the Margulis lemma, we decomposeM into its thick and thin parts. On the compact thick components, we show that Φ M varies from a constant value by no more thanO(\(\sqrt {\lambda (M)}\)). The estimate for λ(M) is calculable in terms of the topology ofM and the lengths of short geodesics ofM. An analogous theorem of the compact case was treated in [SWY].