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Questions tagged [fuchsian-groups]

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

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Critical exponent for groups with parabolics

I'm going to ask this question first in classical setting and then sketch its natural geometric setting. Let $\Gamma$ be a subgroup of $\operatorname{PSL}_2(\mathbb Z)$ (the question is mostly ...
Jean Raimbault's user avatar
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Maximal orders and surface subgroups of even genus

Let $A$ be a quaternion algebra over a totally real number field $k$. Suppose that $A$ splits at exactly one real place of $A$. Let $\mathcal{O}$ be a maximal order in $A$. Then $\mathcal{O}$ contains ...
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Almost modularity of Belyi curves and etale fundamental group of non-Belyi curves

Belyi's theorem states that every curve defined over $\mathbb{\bar Q}$ is almost modular (obtained from $\mathbb{H}^2/\Gamma,\ \Gamma$ a finite index subgroup of $PSL(2,\mathbb{Z})$), after ...
user6419's user avatar
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Regarding fundamental domain of 2 genus surface

Let $\mathbb{H}^2$ be the hyperbolic plane with $(2,3,7)$ tiling. Let $\Gamma$ be a subgroup of $(2,3,7)$ triangle group such that $\mathbb{H}^2/\Gamma$ is the compact orientable surface of genus 2 ...
KAK's user avatar
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Reference for triangle groups

Can anyone suggest to me some references for studying triangle groups? Especially the existence of finite index subgroups, subgroups isomorphic to fundamental groups of compact surfaces etc.
KAK's user avatar
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Area of fundamental domain of Fuchsian group and index of a Fuchsian group in the triangle group

Let $\mathbb{H}$ be the upper half plane model of hyperbolic geometry. Let $\Gamma$ be the Fuchsian group such that $\mathbb{H}/\Gamma$ is the compact orientable surface of genus $2$. Suppose $\Gamma =...
KAK's user avatar
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Classification of fundamental domains of a fuchsian group

Let $G$ be the (2,3,7) triangle group. We can see it as symmetry group of (2,3,7) tiling of the hyperbolic plane or symmetry group of $[3^7]$ tiling of the hyperbolic plane. This contains translations,...
KAK's user avatar
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1 answer
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Analytic continuation of the Eisenstein series defined over Hecke and Fricke subgroups

It is well known that the (real analytic) Eisenstein series is defined, in the slash notation, as follows $$E_{s}(\tau) = \sum\limits_{\gamma\in\Gamma_{\infty}\backslash\text{SL}(2,\mathbb{Z})}\left.y^...
Spoilt Milk's user avatar
2 votes
1 answer
204 views

Examples of group families with solvable uniform word problem

I would like to know of any examples of families of groups that are known (or conjectured) to have a solvable uniform word problem, i.e. an algorithm that given a presentation $P$ of a group in the ...
Agelos's user avatar
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Abelianizations of arithmetic Fuchsian groups

Let $a,b$ be positive integers with $x^2-ay^2-bz^2+abv^2=0$ having only the zero solution over $\mathbb Z$ and consider the Fuchsian group \begin{equation*} \Gamma=\left\{\begin{bmatrix} k+\sqrt{a}l &...
William of Baskerville's user avatar
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The relation between Wolf's and Teichmüller's parametrization of the Teichmüller space

Let $\mathcal{T}_g$ be the Teichmüller space of Riemannian surface structures on an oriented 2-dimensional manifold of genus $g$. Fix a point $S \in \mathcal{T}_g$. There are two different ways to ...
 V. Rogov's user avatar
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1 answer
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How to construct a group of Möbius transformations 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 Möbius transformations: \begin{gather*} z'=z+1 \\ z'=-\frac{1}{z} \end{gather*} and ...
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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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3 votes
1 answer
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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 ...
Sounak Sinha's user avatar
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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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2 votes
1 answer
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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 ...
user178149's user avatar
26 votes
4 answers
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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-...
Jonah Gaster's user avatar
6 votes
2 answers
330 views

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 ...
user163814's user avatar
9 votes
1 answer
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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 ...
dragoboy's user avatar
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2 answers
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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 ...
Tina's user avatar
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Arithmetic product and sum of limit sets of non-elementary Fuchsian group of second kind

Let $L \subset \mathbb{R}$ be a limit set of a Fuchsian group $\Gamma$. If $\Gamma$ is a non-elementary Fuchsian group of second kind, then $L$ is a Cantor set. For example: $\Gamma= \bigg\langle \...
Markiff's user avatar
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2 answers
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$PSL_2(\mathbb{R})$ representations of free groups

Let $S_{g,n}^b$ denote a surface of genus $g$ with $n$ punctures and $b$ boundary components. Let us assume $\max\{b,n\}\geq 1$. It is then obvious that $S_{g,n}^b$ deformation retracts to a bouquet ...
user470881's user avatar
7 votes
0 answers
280 views

Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?

Question: Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe ...
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5 votes
1 answer
201 views

Normal Fuchsian subgroups

I've been working with Fuchsian groups and from geometrical motivations finding a cocompact normal Fuchsian subgroups of $PSL(2,\mathbb{R})$ would have intresting properties for my research. It is ...
P. W. Maunt's user avatar
3 votes
1 answer
156 views

Reference request: geometric finiteness of Fuchsian groups

My limited knowledge on hyperbolic geometry suggests me that the following proposition should be true (please correct me if I'm wrong): Proposition. The convex core of a complete hyperbolic surface ...
Xin Nie's user avatar
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Spectral theory for Fuchsian groups of the first kind

There are tons of material on the spectral theory of $L^2(\Gamma\backslash G)$ for a lattice $\Gamma$ in $G=PSL_2({\mathbb R})$. There are also many papers on the case of $\Gamma$ being convex-...
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2 votes
1 answer
268 views

Build a Fuchsian group starting from punctures on a disk

Consider the moduli space of hyperbolic metrics on the disk with $n>3$ marked points on its boundary, $\mathcal{M}_{D,n}$. $\mathcal{M}_{D,n}$ can be parametrised in terms of cross ratios of the ...
giulio bullsaver's user avatar
3 votes
1 answer
763 views

Hyperbolic Metric on a Riemann Surface

From uniformization theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ punctures such that $2g+n\ge 3$ contains a unique hyperbolic metric. The ...
QGravity's user avatar
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3 votes
2 answers
158 views

What is the homeomorphism from $\Gamma \backslash T_1 \mathbb{H}$ to $T_1(\Gamma \backslash \mathbb{H})$

Let $\mathbb{H}$ be hyperbolic plane, $\Gamma$ is a discrete subgroup of $PSL_2(\mathbb{R}$) so that $\Gamma \backslash \mathbb{H}$ is a compact hyperbolic surface. Maybe it will be very simple to you ...
Skid Row's user avatar
4 votes
1 answer
283 views

Fuchsian group which is derived from a division quaternion algebra, Mixing flows on the quotient space

Suppose a Fuchsian group $\Gamma$ is derived from a division quaternion algebra. Then the quotient space $\Gamma\backslash \mathcal{H}$ is compact. I am reading the book "Fuchsian Groups" of ...
Markiff's user avatar
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30 votes
7 answers
5k views

Why are Fuchsian groups interesting?

I keep hearing that fuchsian groups are interesting for other reasons than the Fuchsian model for hyperbolic Riemann surfaces. What are those reasons? Are the Fuchsian groups with fixed points ...
JS.'s user avatar
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2 answers
317 views

Reference for 'Normal Subgroups of Fuchsian Groups'

I am looking for a reference on how to explicitly construct normal subgroups of a given Fuchsian group. I appreciate any help.
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example of fuchsian groups acting on 2-sphere by G. Martin

Currently I am reading a paper "Infinite group actions on spheres" by Gaven Martin. I am a first year graduate students and I got lots of questions, so one of them is about the following example: (...
Kerr's user avatar
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What is the metric on the Fuchsian model? [closed]

Let $\mathbb{H}$ be the upper half plane, and $\Gamma < SL(2, \mathbb{R})$ be a Fuchsian group. How is the distance between any two points $x, y \in \mathbb{H} / \Gamma$ in the Fuchsian model ...
user avatar
3 votes
0 answers
178 views

One question about iteration on groups

Let $G$ be a finitely generated group, $H$ a subgroup of $G$ of index $n$, with $a_i$ a set of coset representatives and $$G=\displaystyle\bigcup_{i=1}^nH{a_i}.$$ Let $\phi:H\rightarrow G$ be a ...
Grub's user avatar
  • 333
20 votes
1 answer
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Canonical immersion of the double torus

It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the ...
Jjm's user avatar
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12 votes
3 answers
743 views

Quotient of the hyperbolic plane with respect to commutator group of $\pi_1(\Sigma_g)$

Let $\Sigma_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi_1(\Sigma_g)$. Let $\pi\colon \mathbb{H}\to \Sigma_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$? ...
Mohammad Farajzadeh-Tehrani's user avatar
7 votes
1 answer
447 views

Can Galois conjugates of lattices in SL(2,R) be discrete?

Let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$. Suppose that the trace field of $\Gamma$ is a totally real number field of degree $d$. This gives $d$ homomorphisms $\rho_i:\Gamma\to SL(2,\mathbb{R})$ ...
Alex's user avatar
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3 votes
1 answer
486 views

The Fuchsian monodromy problem

I want to understand the argument being made from equation 6.1 to 6.5 in this paper between pages 27-28 6.2, 6.4 and 6.5 are completely out-of-the-blue to me and I have no clue as to from where they ...
user6818's user avatar
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3 votes
0 answers
340 views

Discussion of specific arithmetic triangle groups?

Arithmetic triangle groups were classified in Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan Volume 29, Number 1 (1977), 91-106. The (2,3,7) case was discussed in detail in a number of ...
Mikhail Katz's user avatar
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3 votes
1 answer
138 views

Classification of maximal nonuniform Fuchsian lattices existent?

I am interested in the set of all non-cocompact Fuchsian lattices which all have a distinguished point as cusp, say $\infty$ in the upper half plane model of the hyperbolic plane. Of course, the ...
Maik Köster's user avatar
3 votes
1 answer
227 views

Arithmetic Fuchsian lattices that are not finite index subgroups of Eichler orders?

Lindenstrauss' proof of AQUE (arithmetic quantum unique ergodicity) assumes that the Fuchsian lattice is an Eichler order or, if I understand it correctly, a finite index subgroup of an Eichler order. ...
Maik Köster's user avatar
5 votes
2 answers
787 views

finite index subgroup of a fuchsian group

Given G, a fuchsian group and a finite sub set A of G. Does there exist a finite index subgroup H in G such that inter section of A with H is empty?
Bidyut Sanki's user avatar
0 votes
1 answer
363 views

Fundamental domain for subgroup of fuchsian Schottky group.

Let G be a Fuchsian Schottky group defined by a possibly infinite set of disjoint halfplanes {C_i}_i. Let F be the fundamental domain obtained by intersecting the complements of the C_i's. If H i a ...
Anna Z's user avatar
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1 vote
1 answer
559 views

Fuchsian groups and their normalizers

Let $\Gamma \leq PSL_2(\mathbb{R})$ be a Fuchsian group. What is the relation between $N(\Gamma) = \{ \alpha \in PSL_2(\mathbb{R}) \mid \alpha \Gamma \alpha^{-1} = \Gamma \}$ and $Aut(\Gamma \...
expmat's user avatar
  • 1,241
0 votes
2 answers
464 views

Fuchsian groups and automorphisms of Riemann surfaces

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(\...
expmat's user avatar
  • 1,241
2 votes
1 answer
341 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 ...
Giovanni Moreno's user avatar
4 votes
0 answers
173 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 $\omega = e^{2 \pi i / 3}$. ...
Marty's user avatar
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1 vote
2 answers
787 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_\...
Will Chen's user avatar
  • 10k
9 votes
1 answer
597 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 ...
David Loeffler's user avatar