Questions tagged [fuchsian-groups]

In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R)

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How to construct a Mobius group corresponding to a given fundamental triangle?

Most introductory textbooks on the modular group begin with an introduction of it as the group generated by the two Mobius transformations: $$z'=z+1$$ $$z'=-\frac{1}{z}$$ and immediately after that, ...
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Variation of the geometry of a Dirichlet region as the defining point varies

Let $\Gamma$ a Fuchsian group acting on the hyperbolic plane $\mathfrak{H}$. For me, I am most interested in the case where $\Gamma$ has a fundamental domain that is a finite-polygon with all ...
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Schwarzian derivative, accessory parameters, projective connections

I am looking at the following Riemann surface (let's call it $M$), \begin{equation} y^n=\frac{(x-x_1)(x-x_3)}{(x-x_2)(x-x_4)} \end{equation} which is a Riemann surface of genus $n-1$. It can be ...
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Ideal Ford domain (for finite index subgroup)

Let $G$ be a lattice Fuchsian group with parabolic elements, seen as a discrete subgroup of matrices $g= \begin{pmatrix} \alpha & \overline{\beta} \\ \beta & \overline{\alpha} \end{pmatrix}$...
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Dirichlet domains of cusped hyperbolic surfaces

I'm used to think of cusped hyperbolic surfaces as equipped with ideal (all vertices are ideal) fundamental domain, but this is not in general a Dirichlet domain. Is is true that any such surface has ...
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Dirichlet region of a free group

Let $G$ be a non-uniform lattice Fuchsian group and let $P$ be a Dirichlet region for $G$. In particular $G$ has parabolic elements, $P$ is not compact and has finite area. We are in the unit disc. Is ...
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Equations defining hyperbolic geodesics in $\mathbb C \setminus\{0,1\}$

Let $X=\mathbb C\setminus\{0,1\}$, equipped with the hyperbolic structure it inherits from Klein's modular $\lambda$ function $\lambda:\mathbb H \to X$. In each (non-peripheral and nontrivial) free-...
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Positive genus Fuchsian groups

Let $G$ be a lattice in $SL(2,\mathbb{R})$. Is it always true that there exists a finite index subgroup $F$ of $G$ such that the quotient surface $\mathbb{H}/F$ has positive genus? Is the statement ...
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Fuchsian groups and Eichler's result

Let $G$ be a Fuchsian group of first kind contained in $\text{PSL}_2(\mathbb{R})$. A result of Eichler says, there exists a finite set $S\subset G$ such that any $\gamma$ in $G$ can be written as a ...
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Reference for openness of subspace of PSL(2,R) representation variety corresponding to Teichmüller space

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Hom{Hom}$Let $S$ be a compact oriented surface with nonempty boundary. There are two variants of Teichmuller space for $S$ you might consider: The ...
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Let $\Gamma \subseteq PSL_2(\mathbb{R})$ be a Fuchsian group, possibly containing elliptic elements. Is it true that $N(\Gamma) / \Gamma$, where $N(\Gamma)$ the normalizer of $\Gamma$ in $PSL_2(\... 1answer 300 views Can the finiteness of a Burnside group with two generators be checked algorithmically by using Fuchsian von Dyck groups? BIG EDIT of the previous question "Coverings of the free Burnside groups", never answered. In the paper http://link.springer.com/article/10.1007/BF00046586 (last section) there is an interesting ... 0answers 171 views Asymptotics of arithmetic Fuchsian groups and Shimura curves. I'm interested in what is known/expected about some families of arithmetic Fuchsian groups. Here is the simplest family that I'm interested in: Let$E = Z[\omega]$, where$\omega = e^{2 \pi i / 3}$. ... 2answers 719 views How do you find the genus of a Fuchsian group derived from a quaternion algebra? Let$G$be a Fuchsian group with normalizer$N(G)$inside$PSL(2,13)$Due to the Hurwitz formula, it suffices to find a presentation of$G$of the form: $$\langle x_1,\ldots,x_r,a_1,b_1,\ldots,a_\... 1answer 505 views Are any two Dirichlet domains for a Fuchsian group "comparable"? Let \Gamma be a [EDIT: finitely generated] Fuchsian group of the first kind (i.e. a discrete subgroup of PSL_2(\mathbf{R}) acting on the upper half-plane admitting a fundamental domain of finite ... 1answer 483 views Non congruence subgroups containing congruence subgroups. Does there exist Fuchsian groups, which is not conjugated in SL(2, \mathbb{R}) to a subgroup of SL(2, \mathbb{Z}), but still contains a congruence subgroup? 4answers 480 views Growth of smallest closed geodesic in congruence subgroups? Let \Gamma be one of the classical congruence subgroups \Gamma_0(N), \Gamma_1(N) and \Gamma(N) of SL(2, \mathbb{Z}). How does the lower bound for the length of primitive geodesics on \... 1answer 348 views Genus of arithmetic surface groups It is known that for each genus, only finitely many points in the moduli space of hyperbolic genus g surfaces are arithmetic. I'm wondering if an existence result is known: for which g do we have ... 2answers 463 views Arithmetic Fuchsian group I have the following questions: Are all Fuchsian groups of signature (0;2,2,2,\infty) arithmetic? What is known about the trace fields of these groups? Best, K. 0answers 257 views Is the absolute value of the j-invariant bounded from below on an annulus Let j:\mathbf{H}\to \mathbf{C} be the j-invariant. It's a modular function for \Gamma(1) = \textrm{PSL}_2(\mathbf{Z}). For \epsilon>0 small, let B(\epsilon) be the image of the strip$$\{... 1answer 761 views The smallest positive eigenvalue and the length of the shortest geodesic I'm confused about some things concerning lengths of geodesics on Riemann surfaces and positive eigenvalues of the Laplacian. Moreover, I'm also interested in the relation between these two. Let$X$... 1answer 1k views How nice are representation varieties of Fuchsian groups? Background Let$S_{g,n}$be an oriented surface of genus$g$, with$n$punctures. We explicitly prohibit the non-hyperbolic cases:$g=0$,$n=0,1,2$.$g=1$,$n=0$. Let$\Gamma$be the fundamental ... 2answers 1k views Cusp width for an arbitraty Fuchsian group In Shimura's Intro to Arithmetic Theory of Automorphic Forms, he defines a cusp of a Fuchsian group$\Gamma$as a point$s \in \mathbb{R} \cup \{ \infty \}$that is fixed by a parabolic element of$\...
I apologize in advance if this question is so trivial or too low level. Let $\Gamma$ be a Fuchsian group. Let $\mathcal{F}$ be the set of pairs $(\mu,f)$, where $\mu \in L^\infty(\mathbb{C})$ such ...
Dear all, once again my question is all about $SL_2(\mathbb{Z})$ and $SL_2(\mathbb{Q})$ ! Which elements in $M \in SL_2(\mathbb{Q})$ can you write in the following form: $M= NBN^{-1}$ with \$N \in ...