Modular symbols for Teichmüller curves

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Curtis T. McMullen
Keyword(s):  
Riemann Surface ◽  
Regular Polygon ◽  
Modular Symbols ◽  
Periodic Trajectories ◽  
Polygonal Billiards ◽  
Teichmuller Curves ◽  
Hyperbolic Riemann Surface ◽  
Teichmüller Curves

Abstract 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} .

scholarly journals Examples of malformed subsets of a Riemann surface

1983 ◽  
Vol 24 (2) ◽  
pp. 101-106
Author(s):  
Moses Glasner
Keyword(s):  
Riemann Surface ◽  
Maximum Principle ◽  
Open Subset ◽  
Linear Mapping ◽  
The Maximum Principle ◽  
Hyperbolic Riemann Surface

Let R be a hyperbolic Riemann surface and W an open subset of R with ∂W piecewise analytic. Denote by the space of Dirichlet finite Tonelli functions on R and by π the harmonic projection of . Consider the relative HD–class on W, HD(W;∂W) = {u∈ │ u │ W∈HD(W) and u │ R\W = 0}. The extremization operation μis the linear mapping of HD(W;∂W) into HD(R) defined by μ. Since π preserves values of functions at the Royden harmonic boundary, the maximum principle implies that μis an order preserving injection and that Mμ is an isometry with respect to the supremum norms.

scholarly journals ORBIFOLD POINTS ON PRYM–TEICHMÜLLER CURVES IN GENUS 

2017 ◽  
Vol 18 (4) ◽  
pp. 673-706 ◽  
Author(s):  
David Torres-Teigell ◽  
Jonathan Zachhuber
Keyword(s):  
Elliptic Curves ◽  
Explicit Construction ◽  
Side Product ◽  
Geometric Description ◽  
Flat Surfaces ◽  
Teichmuller Curves ◽  
Teichmüller Curves

For each discriminant $D>1$, McMullen constructed the Prym–Teichmüller curves $W_{D}(4)$ and $W_{D}(6)$ in ${\mathcal{M}}_{3}$ and ${\mathcal{M}}_{4}$, which constitute one of the few known infinite families of geometrically primitive Teichmüller curves. In the present paper, we determine for each $D$ the number and type of orbifold points on $W_{D}(6)$. These results, together with a previous result of the two authors in the genus $3$ case and with results of Lanneau–Nguyen and Möller, complete the topological characterisation of all Prym–Teichmüller curves and determine their genus. The study of orbifold points relies on the analysis of intersections of $W_{D}(6)$ with certain families of genus $4$ curves with extra automorphisms. As a side product of this study, we give an explicit construction of such families and describe their Prym–Torelli images, which turn out to be isomorphic to certain products of elliptic curves. We also give a geometric description of the flat surfaces associated to these families and describe the asymptotics of the genus of $W_{D}(6)$ for large $D$.

scholarly journals Analytic Functions on Some Riemann Surfaces

1963 ◽  
Vol 22 ◽  
pp. 211-217 ◽  
Author(s):  
Nobushige Toda ◽  
Kikuji Matsumoto
Keyword(s):  
Riemann Surface ◽  
Compact Subset ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Open Riemann Surface ◽  
Interesting Theorem ◽  
Hyperbolic Riemann Surface

Some years ago, Kuramochi gave in his paper [5] a very interesting theorem, which can be stated as follows.THEOREM OF KURAMOCHI. Let R be a hyperbolic Riemann surface of the class Of OHR(OHD,resp.). Then, for any compact subset K of R such that R—K is connected, R—K as an open Riemann surface belongs to the class 0AB(OAD resp.).

scholarly journals Limits of canonical forms on towers of Riemann surfaces

2020 ◽  
Vol 2020 (764) ◽  
pp. 287-304
Author(s):  
Hyungryul Baik ◽  
Farbod Shokrieh ◽  
Chenxi Wu
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Type Theorem ◽  
Universal Cover ◽  
Canonical Forms ◽  
Classical Version ◽  
Galois Cover ◽  
Hyperbolic Form ◽  
Galois Covers ◽  
Hyperbolic Riemann Surface

AbstractWe 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.

scholarly journals Adiabatic limit of the eta invariant over cofinite quotients of PSL(2, ℝ)

2008 ◽  
Vol 144 (6) ◽  
pp. 1593-1616 ◽  
Author(s):  
Paul Loya ◽  
Sergiu Moroianu ◽  
Jinsung Park
Keyword(s):  
Riemann Surface ◽  
Dirac Operator ◽  
Continuous Spectrum ◽  
Spin Structure ◽  
Adiabatic Limit ◽  
Regularized Trace ◽  
Standard Definition ◽  
Eta Invariant ◽  
Hyperbolic Riemann Surface

AbstractThe eta invariant of the Dirac operator over a non-compact cofinite quotient of PSL(2,ℝ) is defined through a regularized trace following Melrose. It reduces to the standard definition in terms of eigenvalues in the case of a totally non-trivial spin structure. When the S1-fibers are rescaled, the metric becomes of non-exact fibered-cusp type near the ends. We completely describe the continuous spectrum of the Dirac operator with respect to the rescaled metric and its dependence on the spin structure, and show that the adiabatic limit of the eta invariant is essentially the volume of the base hyperbolic Riemann surface with cusps, extending some of the results of Seade and Steer.

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