Problem 2
Question
Show: (a) The group \(\operatorname{SL}(2, \mathbb{R})\) acts transitively on the upper half-plane H, i.e. for any two points \(z, w \in\) H there exists \(M \in \mathrm{SL}(2, \mathbb{R})\) with \(w=M z\). Hint. It is enough to consider the case \(w=\mathrm{i}\). Then we can even take \(c=0\). (b) The map $$ \begin{array}{r} \mathrm{SL}(2, \mathbb{R}) / \mathrm{SO}(2, \mathbb{R}) \longrightarrow \mathbb{H I} \\ M \cdot \mathrm{SO}(2, \mathbb{R}) \longrightarrow M \mathrm{i} \end{array} $$ is bijective. (It is even a homeomorphism, if we equip its domain of definition with the quotient topology.)
Step-by-Step Solution
Verified Answer
(a) Use matrix \( \begin{pmatrix} 1/y & -x/y \\ 0 & 1 \end{pmatrix} \) to map any \( z \) to \( i \). (b) The map is bijective; each coset corresponds uniquely to a point in \( \mathbb{H} \).
1Step 1: Define the Upper Half-Plane
The upper half-plane \( \mathbb{H} \) is defined as the set of complex numbers \( z = x + yi \) where the real part \( x \) is in \( \mathbb{R} \) and the imaginary part \( y > 0 \).
2Step 2: Define SL(2, R)
The group \( \operatorname{SL}(2, \mathbb{R}) \) consists of all \( 2 \times 2 \) real matrices with determinant 1: $$ M = \begin{pmatrix} a & b \ c & d \end{pmatrix}, \text{ where } ad - bc = 1. $$
3Step 3: Understand the Action of SL(2, R) on H
\( \operatorname{SL}(2, \mathbb{R}) \) acts on \( \mathbb{H} \) by Möbius transformations: for \( z \in \mathbb{H} \) and \( M = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the action is given by \( Mz = \frac{az + b}{cz + d} \).
4Step 4: Prove Transitivity (Part a)
Given any \( z \in \mathbb{H} \), we want to find \( M \in \operatorname{SL}(2, \mathbb{R}) \) such that \( Mz = i \). Assume \( z = x + yi \). Take the matrix \( M = \begin{pmatrix} \frac{1}{y} & -\frac{x}{y} \ 0 & 1 \end{pmatrix} \). Then, \( Mz = \frac{(1/y)(x + yi) - (x/y)}{1} = i \). Since this operation is possible for any \( z \in \mathbb{H} \), the action is transitive.
5Step 5: Define SO(2, R) Subgroup
\( \mathrm{SO}(2, \mathbb{R}) \) is the group of rotations in 2D: matrices of the form \( \begin{pmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix} \) with determinant 1. It is a subgroup of \( \operatorname{SL}(2, \mathbb{R}) \).
6Step 6: Describe the Mapping (Part b)
Consider the map from the coset space \( \mathrm{SL}(2, \mathbb{R}) / \mathrm{SO}(2, \mathbb{R}) \) to \( \mathbb{H} \) by sending a coset \( M \cdot \mathrm{SO}(2, \mathbb{R}) \) to \( Mi \). This map assigns each coset a unique point in \( \mathbb{H} \).
7Step 7: Show Bijectivity
To prove the map is bijective, we show it is both injective and surjective: - **Injective**: If \( Mi = Ni \), then \( M^{-1}N \) must stabilize \( i \), meaning \( M^{-1}N \in \mathrm{SO}(2, \mathbb{R}) \). Hence, \( M = N \cdot \mathrm{SO}(2, \mathbb{R}) \). - **Surjective**: For any \( z \in \mathbb{H} \), we can find some \( M \in \operatorname{SL}(2, \mathbb{R}) \) that maps \( i \) to \( z \), ensuring every point in \( \mathbb{H} \) is reached.
Key Concepts
Upper Half-PlaneMöbius TransformationsTransitive Group ActionSO(2,R)Bijective Mapping
Upper Half-Plane
The Upper Half-Plane, denoted as \( \mathbb{H} \), is an essential concept in complex analysis and various branches of mathematics. It consists of all complex numbers \( z = x + yi \) where the real part \( x \) belongs to \( \mathbb{R} \), the set of real numbers, and the imaginary part \( y \) is greater than zero.
This definition means that you're working with all complex numbers that lie above the real line on the complex plane.
This half-plane serves as a playground for many mathematical operations and transformations, particularly in the field of Möbius transformations and group actions.
This definition means that you're working with all complex numbers that lie above the real line on the complex plane.
This half-plane serves as a playground for many mathematical operations and transformations, particularly in the field of Möbius transformations and group actions.
Möbius Transformations
Möbius transformations are a fascinating type of function that can be used to map the complex plane in various ways, maintaining the structure of lines and circles.
If you have a 2x2 matrix \( M = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) with a determinant of 1, from the group \( \operatorname{SL}(2, \mathbb{R}) \), the transformation is given by:
\[ Mz = \frac{az + b}{cz + d} \] where \( z \) belongs to the upper half-plane.
These transformations are pivotal because they provide an action of the linear group on the upper half-plane, nicely preserving angles and orientations.
If you have a 2x2 matrix \( M = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) with a determinant of 1, from the group \( \operatorname{SL}(2, \mathbb{R}) \), the transformation is given by:
\[ Mz = \frac{az + b}{cz + d} \] where \( z \) belongs to the upper half-plane.
These transformations are pivotal because they provide an action of the linear group on the upper half-plane, nicely preserving angles and orientations.
Transitive Group Action
A transitive group action is when a group acts on a set and for any two elements in the set, you can find a group element that maps one to the other.
In the case of the group \( \operatorname{SL}(2, \mathbb{R}) \) acting on the upper half-plane \( \mathbb{H} \), this means that for any two points \( z, w \in \mathbb{H} \), there is a matrix \( M \) such that \( Mz = w \).
Transitivity in this sense implies a sort of "movement power" over the set; any point can theoretically "reach" any other point through the group action.
It’s important to note that this capability establishes the set \( \mathbb{H} \) as a single orbit under the action of \( \operatorname{SL}(2, \mathbb{R}) \).
In the case of the group \( \operatorname{SL}(2, \mathbb{R}) \) acting on the upper half-plane \( \mathbb{H} \), this means that for any two points \( z, w \in \mathbb{H} \), there is a matrix \( M \) such that \( Mz = w \).
Transitivity in this sense implies a sort of "movement power" over the set; any point can theoretically "reach" any other point through the group action.
It’s important to note that this capability establishes the set \( \mathbb{H} \) as a single orbit under the action of \( \operatorname{SL}(2, \mathbb{R}) \).
SO(2,R)
The group \( \mathrm{SO}(2, \mathbb{R}) \) is known as the Special Orthogonal Group in two dimensions.
It includes all rotation matrices of the form:
\[ \begin{pmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix} \]
where \( \theta \) is a real number. These matrices have a determinant of 1, which maintains the area under transformations, and describe rotations about the origin in the 2-dimensional plane.
\( \mathrm{SO}(2, \mathbb{R}) \) as a subgroup plays a critical role in many symmetry and geometry considerations, particularly as a stabilizer of points in transformation actions.
It includes all rotation matrices of the form:
\[ \begin{pmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix} \]
where \( \theta \) is a real number. These matrices have a determinant of 1, which maintains the area under transformations, and describe rotations about the origin in the 2-dimensional plane.
\( \mathrm{SO}(2, \mathbb{R}) \) as a subgroup plays a critical role in many symmetry and geometry considerations, particularly as a stabilizer of points in transformation actions.
Bijective Mapping
A bijective mapping, quite simply, is a one-to-one correspondence between two sets.
For a function or map to be bijective, it needs both injectivity and surjectivity:
In this exercise, the map from \( \mathrm{SL}(2, \mathbb{R}) / \mathrm{SO}(2, \mathbb{R}) \) to the upper half-plane assigns a unique point to each coset, demonstrating both qualities, thus making it bijective.
Understanding bijections is crucial in many fields of mathematics, helping us understand equivalencies and mappings between seemingly different spaces.
For a function or map to be bijective, it needs both injectivity and surjectivity:
- *Injective (One-to-One)*: Each element of the domain maps to a unique element of the codomain, ensuring no two domain elements share the same map.
- *Surjective (Onto)*: Every element of the codomain has at least one element of the domain pointing to it, ensuring the entire codomain is covered.
In this exercise, the map from \( \mathrm{SL}(2, \mathbb{R}) / \mathrm{SO}(2, \mathbb{R}) \) to the upper half-plane assigns a unique point to each coset, demonstrating both qualities, thus making it bijective.
Understanding bijections is crucial in many fields of mathematics, helping us understand equivalencies and mappings between seemingly different spaces.
Other exercises in this chapter
Problem 1
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The natural group homomorphism $$ \mathrm{SL}(2, \mathbb{Z}) \longrightarrow \mathrm{SL}(2, \mathbb{Z} / q \mathbb{Z}) $$ is surjective. In particular, $$ [\Gam
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Let \(M \in \mathrm{SL}(2, \mathbb{R})\), and let \(l\) be an integer with the property \(M^{l} \neq \pm E\). The matrix \(M\) is elliptic, iff \(M^{l}\) is ell
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