# Questions tagged [mobius-transformation]

For questions about the geometry, complex-analytic, and group-theoretic properties of the Möbius transformations (linear fractional transformations) $z \mapsto \frac{az + b}{cz + d}$ of the complex plane, which can be identified with the group $PGL(2, \mathbb{C})$, or certain subgroups thereof.

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### no wandering domain theorem for circles

I am trying to understand the proof of the no wandering domain theorem from Beardon's iterations of rational functions and thought a good start would be to omit the quasiconformal structures part and ...
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### Mobius transformation mapping three specific points to three specific points

I'm having trouble understanding point (30) on page- 155 of Visual Complex Analysis, the following is given: Let $C$ be the unique circle through the points $q,r,s$ in the z-plane, oriented so that ...
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### Automorphism group of unit disk acts transitively?

Is there a quick way to see that $\text{Aut}(D)$, the group of conformal automorphisms, acts transitively on the unit disk $D$? I know that one can equivalently consider the projective special linear ...
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### What are the infinitesimal generators of the Mobius transformation

I understand that the Mobius transformation $$f(t)=\frac{at+b}{ct+d}$$ is isomorphic to $SL(2)$ for $ad-bc=1$. I also know how to get the infinitesimal generators for the $SL(2)$ group. i.e. the trace-...
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### Mechanics of Clifford Algebra valued Möbius Maps

The Möbius maps $$(ax+b)(cx+d)^{-1}$$ Where $a,b,c,d,x \in \mathbb{C}$ are well-understood, but how do the ones where the variable and coefficients are in Clifford Algebras act? Are there any good ...
1 vote
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### Resources/explanations for conformal geometry with “null cones at infinity”

In the Wikipedia article on conformal geometry https://en.m.wikipedia.org/wiki/Conformal_geometry there’s a section in Mobius geometry that says it’s the study of pseudo Euclidean spaces with either a ...
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### Formula for action of $\operatorname{SL}(2,\mathbb{C})$ on hyperbolic 3-space [duplicate]

It's pretty standard in 3-manifold topology and hyperbolic geometry that $\operatorname{PSL}(2,\mathbb{C})$ is the orientation-preserving isometry group of hyperbolic 3-space $\mathbb H^3$. I haven't ...
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### Law of cosines in proving invariance of Mobius energy

I'm currently reading Freedman's paper on the Mobius invariance of knot energy, and I'm stuck on a particular equality (2.8). Let $\gamma$ be a curve in $\mathbb{R}^3$ parametrized with respect to arc ...
1 vote
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### Show that the center of the circle of the Mobius transformation $f(z)=i(1-z)/(1+z)$ of $z=k+iy$ is always on the $y$ axis.

How can I show that the center of the circle of the Mobius transformation $f(z)=i(1-z)/(1+z)$ of $z=k+iy$ is always on the $y$ axis when $k$ is a constant? I know that this is a direct result of the ...
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### What is the locus of all shape parameters of ideal tetrahedra which share the same volume?

I'm taking a class in hyperbolic knot theory out of Jessica Purcell's book, and I was curious about some volumes and classifications of ideal tetrahedra in $\mathbb{H}^3$, with the upper half $n$-...
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### Find the image of the upper half-plane Im(z) > 0 under the transformation $w = \frac{(1-i)z+2}{(1+i)z+2}$

How can I solve this question? Given that the transformation part is very much similar to $f(z) = \frac{az+b}{cz+d}$, the requested transformation is probably a Möbius transformation. I am not ...
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### Analytic maps from lower half plane onto itself.

I know that the bilinear map $f(z)=\frac{az+b}{cz+d},a,b,c,d\in\mathbb{R},ad−bc>0$, maps the upper half plane onto itself. But I wonder what is the map which maps lower half plane onto itself such ...
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### Möbius Transformation that moves zero

Find a Möbius Transformation on complex plane, that moves $|z|<1$ to $|w|>2$ and point (0, 0) to (4, 0). If we don't concern second condition, we can use $w(z) = \frac{2}{z}$. But later we can't ...
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Most introductory textbooks on the modular group begin with an introduction of it as the group generated by the two Mobius transformations: $$z'=z+1$$ $$z'=-\frac{1}{z}$$, and immediately after that, ...