Modular symbols for Teichmüller curves

This paper introduces a space of nonabelian modular symbols 𝒮 ⁢ ( V ) {{\mathcal{S}}(V)} attached to any hyperbolic Riemann surface V, and applies it to obtain new results on polygonal billiards and holomorphic 1-forms. In particular, it shows the scarring behavior of periodic trajectories for billiards in a regular polygon is governed by a countable set of measures homeomorphic to ω ω + 1 {\omega^{\omega}+1} .

Limits of canonical forms on towers of Riemann surfaces

We prove a generalized version of Kazhdan’s theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence {\{S_{n}\rightarrow S\}} of finite Galois covers of a hyperbolic Riemann surface S, converging to the universal cover. The theorem states that the sequence of forms on S inherited from the canonical forms on {S_{n}}’s converges uniformly to (a multiple of) the hyperbolic form. We prove a generalized version of this theorem, where the universal cover is replaced with any infinite Galois cover. Along the way, we also prove a Gauss–Bonnet-type theorem in the context of arbitrary infinite Galois covers.

On Infinitely generated Fuchsian groups of the Loch Ness monster, the Cantor tree and the Blooming Cantor tree

In this paper, for a non-compact Riemman surface S homeomorphic to either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we give a precise description of an infinite set of generators of a Fuchsian group Γ < PSL(2, ℝ), such that the quotient space ℍ/Γ is a hyperbolic Riemann surface homeomorphic to S. For each one of these constructions, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.

Asymptotic conformality of the barycentric extension of quasiconformal maps

We first remark that the complex dilatation of a quasiconformal homeomorphism of a hyperbolic Riemann surface R obtained by the barycentric extension due to Douady-Earle vanishes at any cusp of R. Then we give a new proof, without using the Bers embedding, of a fact that the quasiconformal homeomorphism obtained by the barycentric extension from an integrable Beltrami coefficient on R is asymptotically conformal if R satisfies a certain geometric condition.

ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES

We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface $M$. Actually, we study the zeros of $(Z_{M}H_{M})^{\prime }$, where $Z_{M}$ is the Selberg zeta function and $H_{M}$ is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite volume hyperbolic Riemann surface $M$. Our main results address finiteness of number of zeros of $(Z_{M}H_{M})^{\prime }$ in the half-plane $\operatorname{Re}(s)<1/2$, an asymptotic count for the vertical distribution of zeros, and an asymptotic count for the horizontal distance of zeros. One realization of the spectral analysis of the Laplacian is the location of the zeros of $Z_{M}$, or, equivalently, the zeros of $Z_{M}H_{M}$. Our analysis yields an invariant $A_{M}$ which appears in the vertical and weighted vertical distribution of zeros of $(Z_{M}H_{M})^{\prime }$, and we show that $A_{M}$ has different values for surfaces associated to two topologically equivalent yet different arithmetically defined Fuchsian groups. We view this aspect of our main theorem as indicating the existence of further spectral phenomena which provides an additional refinement within the set of arithmetically defined Fuchsian groups.

A uniform bound for geodesic periods of eigenfunctions on hyperbolic surfaces

We consider periods along closed geodesics and along geodesic circles for eigenfunctions of the Laplace–Beltrami operator on a compact hyperbolic Riemann surface. We obtain uniform bounds for such periods as the corresponding eigenvalue tends to infinity. We use methods from the theory of automorphic functions and, in particular, the uniqueness of the corresponding invariant functionals on irreducible unitary representations of PGL