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Questions tagged [modular-group]

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0answers
51 views

Modulo Arithmetic hidden property (a mod n+ 1)

I was solving a programming question where we need to find compact form of a number and finding difficult to understand hidden modulo property behind it . I searched a lot but could not figure out. ...
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1answer
62 views

Free subgroups of $PSL(2,\mathbb{Z})$ of index 6

There are two "natural" subgroups of $PSL(2,\mathbb{Z})\cong C_2\ast C_3$ of index 6. One is the congruence subgroup $\Gamma_0(2)$ which is the kernel of the map $PSL(2,\mathbb{Z})\to PSL(2,\mathbb{Z}/...
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Does every subgroup of finite index contain a power of each element of the group?

Let $G$ be a group, not necessarily finite. If $H$ is a normal subgroup of $G$ of a finite index, say $(G:H)=n$, then for every $g\in G$ we have $g^n\in H$. Does this statement remain valid if do not ...
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0answers
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Equality of lattices

Let $H$ be the upper half plane and for $\tau\in H$ let $\Lambda_\tau=\tau\mathbb Z+\mathbb Z$. We know that the lattices $\Lambda_\tau$ and $\Lambda_\omega$ are homothetic if and only if $\tau\equiv ...
2
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1answer
48 views

When $J(\tau)\in\mathbb{R}$? ($J$ is Klein's $j$-invariant.)

I would like to know the set $X=J^{-1}(\mathbb{R})$, where $J$ is Klein's $J$-invariant. Since $J$ is a modular function, it suffices to know the intersection of $X$ and the fundamental domain $\{\tau\...
2
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1answer
83 views

Why is this discrete subgroup of $PSL(2,\mathbb{C})$ not Kleinian?

Definition: For a subgroup $G$ of the group $PSL(2,\mathbb{C})$ acting on $\mathbb{P}^1$, its domain of discontinuity is the set of all points, $z$, with the following properties: $1.$ The ...
2
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1answer
38 views

modular group, prime ideals

I'm probably in over my head, but I came across the following sentence in a thesis by Evan Oliver entitled "Congruence Subarrangements of the Schmidt Arrangement": "The modular group is the set of ...
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1answer
149 views

Name of the modular group

I've been studying the hyperbolic plane and the action of the group $PSL(2,\mathbb{R})$ on it. I found that the modular group $PSL(2,\mathbb{Z})$ is a discrete subgroup of $PSL(2,\mathbb{R})$ so it's ...
3
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1answer
108 views

Action of full modular group on higher level modular functions

Let $\Gamma$ be a finite index subgroup of $\Gamma(1)=SL_2(\mathbf{Z})$ and $f$ a modular function for $\Gamma$. By this I mean a meromorphic function defined on the upper half-plane $f: \mathfrak{h} \...
3
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1answer
113 views

Decomposition of modular group elements

The modular group $PSL_2(\mathbb{Z})$ acts on the hyperbolic half-space $H$ by $$h\cdot z=\frac{az+b}{cz+d},\;z\in H,\;h=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in PSL_2(\mathbb{Z})$$ with $ad-bc=...
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3answers
361 views

The commutator subgroup of modular group is a free group of rank 2

In the paper http://projecteuclid.org/download/pdf_1/euclid.ijm/1255632506 it is stated without proof that the commutator subgroup of the modular group is a free group of rank $2$. Can anyone give a ...
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1answer
352 views

Elliptic Points of Modular Group in Upper Half Plane

This is a very small question. Let $\mathbb{\Gamma} = \mathrm{SL_2}(\mathbb{Z})$ be the modular group, $\mathcal{F} = \{z \in \mathbb{C} ;\; \lvert z \rvert \geq 1,\; \lvert \Re (z) \rvert \leq 1/2\}...
2
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0answers
141 views

Modular group representation

Does anyone know how to describe Möbius transformations with integer coefficients defined on the upper half plane in terms of $z+1$ and $1/z$? Some people call it the modular group. I would ...
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117 views

generalisations of the modular group

I read about how Hecke groups are a particular generalisation of the modular group, generalising one of the generators of the modular group to $z\mapsto z + \lambda$. Have people studied ...