The fuchsian-groups tag has no usage guidance.

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### 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 ...

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145 views

### 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 ...

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420 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 $\mathbb{H}/[G,G]$?
...

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274 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 $\rho_i:\Gamma\to SL(2,\mathbb{R})$ ...

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**1**answer

349 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 ...

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182 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 ...

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123 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 ...

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**1**answer

149 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. ...

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352 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?

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176 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 ...

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**1**answer

200 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 ...

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196 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 ...

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**1**answer

177 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 ...

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138 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 $\omega = e^{2 \pi i / 3}$. ...

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478 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 ...

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**1**answer

323 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 ...

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**1**answer

374 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?

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413 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 ...

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**1**answer

231 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 ...

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304 views

### Arithmetic Fuchsian group

Dear all,
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.

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217 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 ...

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448 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 $X$ ...

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454 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 ...

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554 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 ...

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**1**answer

307 views

### Conjugate Groups of (quasi) Fuchsian Groups

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 ...

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378 views

### Which elements in SL2(Q) are conjugated to an element in SL2(Z)

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 ...

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699 views

### Has a conjugation of SL2(Z) finite index in SL2(Z)? (Modular group)

Dear all,
I have a probably rather simple question: Suppose we have a Matrix $ M\in SL_2(\mathbb{Q}) $. Does the group $ M^{-1} SL_2(\mathbb{Z}) M \cap SL_2(\mathbb{Z})$ then always have finite index ...

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415 views

### Triangle groups - uniqueness and trace field

Dear all,
again I need your help for the following two questions: Suppose we have a triangle group of signature (p,q,\infty).
1) When is such a group unique (up to isomorphism)?
2) Do you have a ...

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855 views

### fundamental domains for free fuchsian group.

I try to understand some of the topology of the space of pointed non-compact hyperbolic surfaces (with the pointed Gromov-Hausdorff topology). It is known that the fundamental
group of a non-compact ...