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In the book Indra's pearls, Möbius transformations are used to construct Kleinian fractals, which are limit sets of a free group generated by two Möbius transformations $a$ and $b$.

In the process of learning, it was very helpful to identify the fundamental domain of such a free group. If we restrict ourselves to just one Möbius transformation $a$ the fundamental domain corresponds to the complex plane with two discs removed. Among other examples they identify the fundamental domain $\mathcal{F}$ for the one-parametric family of M"obius transformations: $$a: z\mapsto {z+\cos\theta\over \cos\theta \cdot z+1}$$ then the fundamental domain is given by: $$\mathcal{F}=\mathbb{C} \setminus \{D_1 \cup D_2\}$$ where the two discs are given by: $$D_1=\{z\in \mathbb{C}: |z-{1\over \cos\theta}|\le \tan\theta\}$$ and $$D_2=\{z\in \mathbb{c}: |z+{1\over \cos\theta}|\le \tan\theta\}$$

The two discs are bounded by the two circles with radius $\tan\theta$ located at $x=\pm {1\over \cos\theta}$.

I want to ask, how one can construct such a fundamental domain for an arbitrary Möbius transformation?

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    $\begingroup$ For a general construction of fundamental domain (of any properly discontinuous group of linear-fractional transformation) you may consult the book Ford, Automorphic functions, or any other book on Kleinian groups. $\endgroup$
    – Alexandre Eremenko
    Commented Feb 3, 2023 at 0:17
  • $\begingroup$ I tried to buy the book by Ford. But I can only find it in the US not shipable to Europe. On the other hand, I found the first 25 pages of the book at googlebooks. Within these 25 pages the isometric circle is discussed, which happens to be exactly the border of the two discs $D_1$ and $D_2$ for the example I showed in the question. Can I generalize this observation, that the fundamental domain is $\mathbb{C}$ with the discs removed that are surrounded by the isometric circles of $a$ and its inverse $a^{-1}$? This would be very neat and simple as well. $\endgroup$
    – p6majo
    Commented Feb 4, 2023 at 19:49
  • $\begingroup$ Some other, more modern books are Maskit, Kleinian groups, and A. Beardon, Geometry of discrete groups. They also contain constructions of fundamental regions. There are two basic constructions: one is called a Ford fundamental region, another Dirichlet fundamental region. $\endgroup$
    – Alexandre Eremenko
    Commented Feb 5, 2023 at 1:19

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