Marked Riemann surfaces and Fuchsian groups and Fenchel's realization theorem

1981 ◽  
pp. 140-203
Author(s):  
Heiner Zieschang
Keyword(s):  
Riemann Surfaces ◽  
Fuchsian Groups

scholarly journals Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces

2019 ◽  
Vol 40 (8) ◽  
pp. 2017-2072
Author(s):  
MAURO ARTIGIANI ◽  
LUCA MARCHESE ◽  
CORINNA ULCIGRAI
Keyword(s):  
Diophantine Approximation ◽  
Riemann Surfaces ◽  
Main Tool ◽  
Height Function ◽  
Classical Case ◽  
Fuchsian Groups ◽  
Proper Function ◽  
Lipschitz Norm ◽  
Lagrange Spectrum ◽  
Hall Ray

We study Lagrange spectra at cusps of finite area Riemann surfaces. These spectra are penetration spectra that describe the asymptotic depths of penetration of geodesics in the cusps. Their study is in particular motivated by Diophantine approximation on Fuchsian groups. In the classical case of the modular surface and classical Diophantine approximation, Hall proved in 1947 that the classical Lagrange spectrum contains a half-line, known as a Hall ray. We generalize this result to the context of Riemann surfaces with cusps and Diophantine approximation on Fuchsian groups. One can measure excursion into a cusp both with respect to a natural height function or, more generally, with respect to any proper function. We prove the existence of a Hall ray for the Lagrange spectrum of any non-cocompact, finite covolume Fuchsian group with respect to any given cusp, both when the penetration is measured by a height function induced by the imaginary part as well as by any proper function close to it with respect to the Lipschitz norm. This shows that Hall rays are stable under (Lipschitz) perturbations. As a main tool, we use the boundary expansion developed by Bowen and Series to code geodesics and produce a geometric continued fraction-like expansion and some of the ideas in Hall’s original argument. A key element in the proof of the results for proper functions is a generalization of Hall’s theorem on the sum of Cantor sets, where we consider functions which are small perturbations in the Lipschitz norm of the sum.

scholarly journals ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF THE MODULI SPACE OF RIEMANN SURFACES OF GENUS 4

2010 ◽  
Vol 52 (2) ◽  
pp. 401-408 ◽  
Author(s):  
ANTONIO F. COSTA ◽  
MILAGROS IZQUIERDO
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Fuchsian Groups ◽  
Branch Locus

AbstractUsing uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space of surfaces of genus 4, we prove its connectedness. As a consequence, one can deform a surface of genus 4 with automorphisms, i.e. symmetric, to any other symmetric genus 4 surface through a path consisting entirely of symmetric surfaces.

Calculation of Fuchsian groups associated to billiards in a rational triangle

1998 ◽  
Vol 18 (4) ◽  
pp. 1019-1042 ◽  
Author(s):  
CLAYTON C. WARD
Keyword(s):  
Algebraic Equation ◽  
Fuchsian Group ◽  
Riemann Surfaces ◽  
Regular Polygon ◽  
Fuchsian Groups ◽  
Geometric Description ◽  
Fermat Curves ◽  
Simply Connected

We define, following Veech, the Fuchsian group $\Gamma(P)$ of a rational polygon $P$. If $P$ is simply-connected, then ‘rational’ is equivalent to the condition that all interior angles of $P$ be rational multiples of $\pi$. Should it happen that $\Gamma(P)$ has finite covolume in $\mathop{\rm PSL}\nolimits (2, {\Bbb R})$ (and is thus a {\it lattice}), then a theorem of Veech states that every billiard path in $P$ is either finite or uniformly distributed in $P$.We consider the Fuchsian groups of various rational triangles. First, we calculate explicitly the Fuchsian groups of a new sequence of triangles, and discover they are lattices. Interestingly, the lattices found are not commensurable with those previously known. We then demonstrate a class of triangles whose Fuchsian groups are {\it not\/} lattices. These are the first examples of such triangles. Finally, we end by showing how one may specify algebraically, i.e. by an explicit polynomial in two variables, the Riemann surfaces and holomorphic one-forms that are associated to a simply-connected rational polygon. Previously, these surfaces were known by their geometric description. As an example, we show a connection between the billiard in a regular polygon and the well-known Fermat curves of the algebraic equation $x^n + y^n = 1$.

Classification of p-groups of automorphisms of Riemann surfaces and their lower central series

1987 ◽  
Vol 29 (2) ◽  
pp. 237-244 ◽  
Author(s):  
Reza Zomorrodian
Keyword(s):  
Nilpotent Group ◽  
Riemann Surfaces ◽  
Lower Central Series ◽  
Nilpotent Groups ◽  
Fuchsian Groups ◽  
Central Series ◽  
Local Form ◽  
Precise Information ◽  
Groups Of Automorphisms ◽  
Hurwitz Theorem

In a previous paper [7], I have made a study of the ”nilpotent” analogue of Hurwitz theorem [4] by considering a particular family of signatures called ”nilpotent admissible” [5]. We saw however, that if μN(g) represents the order of the largest nilpotent group of automorphisms of a surface of genus g < 2, then μN(g) < 16(g − 1) and this upper bound occurs when the covering group is a triangle group having the signature (0; 2,4,8) which is in its own 2-local formThe restriction to the nilpotent groups enabled me to obtain much more precise information than was available in the general case. Moreover, all nilpotent groups attaining this maximum order turned out to be ”2-groups”. Since every finite nilpotent group is the direct product of its Sylow subgroups and the groups of automorphisms are factor groups of the Fuchsian groups, it is natural for us to study the Fuchsian groups havin p-local signatures to obtain more precise information about the finite p-groups, and hence about the finite nilpotent groups.This suggests a new problem of determining for each prime p, the “p-group” analogue of Hurwitz theorem. It turns out, as often happens in questions of this nature, that p = 2 and p = 3 are indeed quite exceptional and harder to deal with while computing their lower central series than other primes. Actually, p = 3 is the most difficult, but all the other primes p ≥ 5 can be dealt with at once.

scholarly journals EQUISYMMETRIC STRATA OF THE MODULI SPACE OF CYCLIC TRIGONAL RIEMANN SURFACES OF GENUS 4

2009 ◽  
Vol 51 (1) ◽  
pp. 19-29 ◽  
Author(s):  
MILAGROS IZQUIERDO ◽  
DANIEL YING
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Fuchsian Groups ◽  
Regular Covering ◽  
Closed Riemann Surface

AbstractA closed Riemann surface which can be realized as a three-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. If the trigonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic trigonal Riemann surface. Using the characterization of cyclic trigonality by Fuchsian groups, we find the structure of the space of cyclic trigonal Riemann surfaces of genus 4.

scholarly journals On extremal Riemann surfaces and their uniformizing fuchsian groups

2002 ◽  
Vol 44 (01) ◽  
pp. 149 ◽  
Author(s):  
Ernesto Girondo ◽  
Gabino González-Diez
Keyword(s):  
Riemann Surfaces ◽  
Fuchsian Groups
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