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Orbits of A, N and K subgroups as cycles and invariant transversal

Exercise I.4.16 [1] $ \newcommand{\scalar}[3][]{\left\langle #2,#3 \right\rangle_{#1}} \newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#1}} \newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#1}} \newcommand{\cycle}[3][]{{#1 C^{#2}_{#3}}} \newcommand{\realline}[3][]{#1 R^{#2}_{#3}} \newcommand{\bs}{\breve{\sigma}} \newcommand{\zcycle}[3][]{#1 Z^{#2}_{#3}} \newcommand{\SL}[1][2]{\mathrm{SL}_{#1}(\Space{R}{})} \newcommand{\rs}{\sigma_r} \newcommand{\lvec}[1]{\overrightarrow{#1}} \newcommand{\rmi}{\mathrm{i}} \newcommand{\alli}{\iota} \newcommand{\rme}{\mathrm{e}} \newcommand{\rmd}{\mathrm{d}} \newcommand{\rmc}{\mathrm{c}} \newcommand{\rmh}{\mathrm{j}} \newcommand{\rmp}{\varepsilon} \newcommand{\modulus}[2][]{\left| #2 \right|_{#1}} \newcommand{\sperp}{⋋} $ Use the similarity formula for the following: these cycles are invariant under the similarity with elements of the

  1. Show that the real axis $v=0$ is represented by the line passing through $(0,0,1,0)$ and a matrix $\begin{pmatrix} s\rmc&0\\ 0&s\rmc \end{pmatrix}$. For any $ \begin{pmatrix} a&b\\ c& d \end{pmatrix}\in \SL$, we have \begin{equation} \begin{pmatrix} a&b\\ c& d \end{pmatrix} \begin{pmatrix} s\rmc&0\\ 0&s\rmc \end{pmatrix} \begin{pmatrix} d & -b\\ -c & a \end{pmatrix} = \begin{pmatrix} s\rmc &0\\ 0&s\rmc \end{pmatrix}, \end{equation} i.e. the real line is $\SL$-invariant.
  2. Write matrices which describe cycles represented by $A$, $N$ and $K$-orbits. Verify that matrices representing

respective subgroups $A$, $N$ and $K$. 3. Show that cycles $(1,0,n,\sigma)$, which are orbits of isotropy groups, are invariant under the respective matrix similarity for the respective values of $\sigma=\alli^2$ and any real $n$. 4. Find the cycles which are transverse to orbits of the isotropy subgroups, i.e.\ are obtained from the vertical axis by the corresponding actions.

Solution. It is straightforward for the real line:

In [1]:
from init_cycle import *
print("The real line is preserved by the moebius maps: %s" % \
real_line.is_equal(real_line.sl2_similarity(a,b,c,d,e)))

            Python wrappers for MoibInv Library
     ---------------------------------------------
Please cite this software as
V.V. Kisil, MoebInv: C++ libraries for manipulations in non-Euclidean geometry, SoftwareX, 11(2020),100385. doi:10.1016/j.softx.2019.100385.
     ---------------------------------------------

Using vector formalism and idx
The real line is preserved by the moebius maps: True
In [2]:
C2=cycle2D(0,[1,n],0,e)
Latex("Cycle $%s$ is A-orbit: %s " % \
(C2.string(), C2.is_equal(C2.sl2_similarity([a,0,0,1/a],e))))
Out[2]:
Cycle $(0, {\left(\begin{array}{cc}1&n\end{array}\right)}^{{symbol139} }, 0)$ is A-orbit: True
In [3]:
C2=cycle2D(0,[0,n],0,e)
Latex("Cycle $%s$ is N-orbit: %s " % \
(C2.string(), C2.is_equal(C2.sl2_similarity([1,b,0,1],e))))
Out[3]:
Cycle $(0, {\left(\begin{array}{cc}0&n\end{array}\right)}^{{symbol169} }, 0)$ is N-orbit: True

Ortits of the isotropy subgroup

In [4]:
C2=cycle2D(1,[0,n],-sign,e) 
Latex("Cycle $%s$ is orbit of isotropy subgroups: %s" % \
(C2.string(), C2.is_equal(C2.sl2_similarity(a,sign*b,b,a,e))))
Out[4]:
Cycle $(1, {\left(\begin{array}{cc}0&n\end{array}\right)}^{{symbol190} }, - \sigma)$ is orbit of isotropy subgroups: True
In [5]:
CV=cycle2D(0, [1,0], 0, e)  
Latex("K-rotation of the vertical axis: $%s$" % \
CV.sl2_similarity(a,-b,b,a,e).normalize().string())
Out[5]:
K-rotation of the vertical axis: $(1, {\left(\begin{array}{cc}\frac{1}{2} \frac{a}{b}-\frac{1}{2} \frac{b}{a}&0\end{array}\right)}_{{symbol272} }, -1)$
In [6]:
Latex("A-rotation of the vertical axis: $%s$" % \
CV.sl2_similarity(a,b,b,a,e).normalize().string())
Out[6]:
A-rotation of the vertical axis: $(1, {\left(\begin{array}{cc}\frac{1}{2} \frac{a}{b}+\frac{1}{2} \frac{b}{a}&0\end{array}\right)}_{{symbol318} }, 1)$
In [7]:
Latex("Np-rotation of the vertical axis: $%s$" % \
CV.sl2_similarity([1,b,0,1],e).string())
Out[7]:
Np-rotation of the vertical axis: $(0, {\left(\begin{array}{cc}1&0\end{array}\right)}_{{symbol346} }, 2 b)$

This notebook is a part of the MoebInv notebooks project [2] .

References

  1. Vladimir V. Kisil. Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of $SL_2(\mathbb{R})$. Imperial College Press, London, 2012. Includes a live DVD.

  2. Vladimir V. Kisil, MoebInv notebooks, 2019.

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