Calculation of the isotropy subgroup orbits¶

Exercise I.5.24 [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{\rmh}{\mathrm{j}} \newcommand{\rmp}{\varepsilon} \newcommand{\modulus}[2][]{\left| #2 \right|_{#1}} \newcommand{\sperp}{⋋} $ Fulfil the following steps:

Write the coefficients of the $\sigma$-zero-radius cycle $\zcycle{}{\sigma}(\alli)$ in $\Space[\sigma]{R}{}$ with e-centre at the hypercomplex unit $\alli=(0,1)$.
According to Exercise ex-centre-zero-moebius.ipynb aka ex.I.5.18.v.ipynb, $\zcycle{}{\sigma}(\alli)$ is invariant under the similarity $\zcycle{}{\sigma}(\alli) \mapsto h\zcycle{}{\sigma}(\alli)h^{-1}$ with $h$ in the respective isotropy subgroups $K$, $N'$ and $A'$ of $\alli$. Check this directly.
Write the coefficients of a generic cycle in the pencil spanned by $\zcycle{}{\sigma}(\alli)$ and the real line. Note that the real line is also invariant under the action of the isotropy subgroups (as any other Möbius transformations) and conclude that any cycle from the pencil will also be invariant under the action of the isotropy subgroups. In other words, those cycles are orbits of the isotropy subgroups. Check that you obtained their equation.

Solution. Define the zero-radius cycle at (0,1)

from init_cycle import *
Zi=Z.subs({u : 0,v : 1})
Latex("Zero-radius cycle with centre at $(0,1)$ is $%s$" % Zi.string())

            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

Check that this cycle is invariant under isotropy sybgroup action

Latex("The cycle is invariant under isotropy subgroup of $(0,1)$: %s" %\
Zi.sl2_similarity(a,sign*b,b,a,e).is_equal(Zi))

Build equation of the orbit as a pencil of the zero-radius cycle
and the real line

Latex("Equation for an isotropy subgroup orbit is $%s$" % \
(Zi.exmul(t)+real_line).normalize().string())

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

References¶

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.
Vladimir V. Kisil, MoebInv notebooks, 2019.

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