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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: (...
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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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...