$123456789 \times x$= What can $x$ be to give an obvious answer? I have got $0,1,10,10^n$ for integer n, $\pi$ and all the above mltiply by $i$ will also be an obvious answer. Are there any other solutions? Any help/contribution appreciated. Note: $\pi$ works because you must give an exact an...
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, perhaps generalising to $z \mapsto \dfrac{-1}{z^n...
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 group come from?How is it related with the Moduli...
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 stabilizer $G_z$ of $z$ is finite. $2.$ $\exists U$, a neighbourhood of $z$ such that, $\space \space\...
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 appreciate any suggestions. Thanks.
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 all matrices in P SL2(Z) whose elements reduce component-wise to the identity matrix over a prime i...
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 $S$ and $T$ such that $$S^2=(ST)^3=1,$$ where we can ...
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 $f: \mathfrak{h} \to \mathbf{C}$, satisfying a growth condition at infinity, which is invariant by $\Gamma$: for e...
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 reference for the proof of this fact?
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 understand the following argument made by Toshits...
Let $\mathbb{K}$ be a field and $A$ a finite dimensional $\mathbb{k}$-algebra with identity $1_{A}$. For two $A$-modules $M$ and $N$ we have the set of all $A$-linear maps from $M$ to $N$ denoted $\mathrm{Hom}_{A}(M,N)$. This set is not in general an $A$-module neither by the left nor by the righ...
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