# Questions tagged [modular-group]

70 questions

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votes

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### Center of $\pi_{1}(\mathbb{R}^{3} \backslash \text{trefoil knot})$ and $\mathrm{PSL}_{2}(\mathbb{Z})$

One of my friend tell me the following question:
What is a center of the group $G = \langle a, b | a^{2} = b^{3}\rangle$?
It is well-known that this is a fundamental group of a complement of a trefoil knot. However, I don't know whether there is…

Seewoo Lee

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**6**

votes

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### Index of a subgroup of the Modular Group

The subgroup of $SL(2,\mathbb{Z})$ generated by
$\begin{pmatrix}1&0\\1&1\end{pmatrix}$ and $\begin{pmatrix}1&5\\0&1\end{pmatrix}$
has come up in a research question in string theory, and I am interested in determining whether or not its index is…

Daniel Longenecker

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**6**

votes

**2**answers

### 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 assume $H$ to be normal?
In particular let…

Shimrod

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votes

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### 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, $\mathcal{F} = \{z \in \mathbb{C} ;\; \lvert z \rvert \geq 1,\; \lvert \Re (z) \rvert \leq 1/2\}$ its fundamental domain.
I (probably) don't…

k.stm

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**4**

votes

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### Decompose elements in $\Gamma_0(N)$

Consider the groups
\begin{align*}
\Gamma_0(N)
\; &:= \;
\biggl\{
\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z})
\;:\;
c \equiv 0 \mod N
\biggr\}
\\
\Gamma_\infty
\; &:= \;
\biggl\{
…

aahlback

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### Proving Diamond & Shurman Exercise 3.7.1, about conjugacy class of $\Gamma_0^{\pm}(N)$.

I am reading chapter 3 of A First Course in Modular Forms but have troubles in Exercise 3.7.1 (c) and (d).
(c) Show that the $\Gamma_0^{\pm}(N)$-conjugacy class of $\gamma \in \Gamma_0(N)$ is the union of the $\Gamma_0(N)$ conjugacy classes of…

Fan Yun Hung

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**4**

votes

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### Explicit equations for $Y(N)$ for small $N$

Consider the congruence subgroup
$$\Gamma(N) = \left\{\left(\begin{array}{cc}
a & b \\
c & d\end{array} \right) \in SL_2(\mathbb{Z})\ ;\ \left(\begin{array}{cc}
a &…

user806056

**4**

votes

**0**answers

### Fundamental domain for congruence subgroup.

Let $\Gamma$ be a congruence subgroup of the modular group $\mathrm{SL}_{2}(\mathbb{Z})$.
Let $R$ be coset representatives of the quotient $\Gamma\setminus\mathrm{SL}_{2}(\mathbb{Z})$ and let
$\mathcal{D}$ be the standard fundamental domain for…

richarddedekind

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**4**

votes

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### $SL(2, \Bbb Z)$ has only one cusp

Let $\Gamma$ be a congruence subgroup of $SL(2, \Bbb Z)$. A cusp is an equivalence of $\Bbb Q\cup\{\infty\}$ under $\Gamma$-action.
What's the meaning of "equivalence $\Bbb Q\cup\{\infty\}$ under $\Gamma$-action?
How to use the above definition to…

mathbeginner

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votes

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### 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 generalisations of the other generator of the modular group,…

user7485

**4**

votes

**1**answer

### 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 Fuchsian.But where does the name of the Modular…

mathman

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**3**

votes

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### 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 $ad-bc=1$. The modular group is generated by two elements $S$…

ArbiterKC

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**3**

votes

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### Irreducible representations of modular group $SL(2,Z)$ with finite image

The classification of irreducible representations of modular group $SL(2,Z)$ is difficult. But for the representations with finite image (the 1-dimensional representation is simple, with only 12), the 2-dimensional case is well established, with…

ReiReiRei

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**3**

votes

**2**answers

### Modified weight 2 Eisenstein series is a modular form for $\Gamma_0(N)$

I'm doing exercise 1.2.8(e) in Diamond & Shurman's A First Course in Modular Forms. The problem is to show that $G_{2,N}(\tau) := G_2(\tau)-NG_2(N\tau)$ is in $M_2(\Gamma_0(N))$. To show this, I need to argue that $G_{2,N}$ satisfies…

Nico

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votes

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### Proving if $\gamma\in SL(2,\mathbb{Z})$ has order $6$, then $\gamma$ is conjugate to $\begin{bmatrix}0 & -1\\ 1 & 1\end{bmatrix}^{\pm1}$

I am reading Diamond's book on modular forms, and I came across the proof that the stabilizers of elliptic points on the upper half-plane are cyclic. To prove that, Diamond starts to show that a transformation that fixes a point must have order $1,…

justadzr

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