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Malik Younsi
Malik Younsi
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Dear Mathoverflow Community,

Suppose that $\Omega$ is a domain in the Riemann Sphere $\widehat{\mathbb{C}}$ with $\infty \in \Omega$, and assume that every connected component of $\partial \Omega$ is a round circle.

Let $\Gamma(\Omega)$ denote the Schottky group of $\Omega$, that is, the free group of Möbius and anti-Möbius transformations generated by the family of reflections across the boundary circles.

Question : Can the limit set $\widehat{\mathbb{C}} \setminus \bigcup_{T \in \Gamma(\Omega)} T(\Omega)$ have positive area.?

Remark : If there are only finitely many boundary circles, then the limit set is a Cantor set and it is not difficult to prove that it must have zero area. We can therefore assume that there are infinitely many boundary circles.

Thank you

Dear Mathoverflow Community,

Suppose that $\Omega$ is a domain in the Riemann Sphere $\widehat{\mathbb{C}}$ with $\infty \in \Omega$, and assume that every connected component of $\partial \Omega$ is a round circle.

Let $\Gamma(\Omega)$ denote the Schottky group of $\Omega$, that is, the free group of Möbius and anti-Möbius transformations generated by the family of reflections across the boundary circles.

Question : Can the limit set $\widehat{\mathbb{C}} \setminus \bigcup_{T \in \Gamma(\Omega)} T(\Omega)$ have positive area.

Remark : If there are only finitely many boundary circles, then the limit set is a Cantor set and it is not difficult to prove that it must have zero area. We can therefore assume that there are infinitely many boundary circles.

Thank you

Dear Mathoverflow Community,

Suppose that $\Omega$ is a domain in the Riemann Sphere $\widehat{\mathbb{C}}$ with $\infty \in \Omega$, and assume that every connected component of $\partial \Omega$ is a round circle.

Let $\Gamma(\Omega)$ denote the Schottky group of $\Omega$, that is, the free group of Möbius and anti-Möbius transformations generated by the family of reflections across the boundary circles.

Question : Can the limit set $\widehat{\mathbb{C}} \setminus \bigcup_{T \in \Gamma(\Omega)} T(\Omega)$ have positive area?

Remark : If there are only finitely many boundary circles, then the limit set is a Cantor set and it is not difficult to prove that it must have zero area. We can therefore assume that there are infinitely many boundary circles.

Thank you

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Malik Younsi
Malik Younsi
  • 2.2k
  • 2
  • 25
  • 44

Dear Mathoverflow Community,

I apologize in advance if the answer to the following question is well-known to experts :

Suppose that $\Omega$ is a domain in the Riemann Sphere $\widehat{\mathbb{C}}$ with $\infty \in \Omega$, and assume that every connected component of $\partial \Omega$ is a round circle.

Let $\Gamma(\Omega)$ denote the Schottky group of $\Omega$, that is, the free group of Möbius and anti-Möbius transformations generated by the family of reflections across the boundary circles.

Question : Can the limit set $\widehat{\mathbb{C}} \setminus \bigcup_{T \in \Gamma(\Omega)} T(\Omega)$ have positive area.

Remark : If there are only finitely many boundary circles, then the limit set is a Cantor set and it is not difficult to prove that it must have zero area. We can therefore assume that there are infinitely many boundary circles.

Thank you

Dear Mathoverflow Community,

I apologize in advance if the answer to the following question is well-known to experts :

Suppose that $\Omega$ is a domain in the Riemann Sphere $\widehat{\mathbb{C}}$ with $\infty \in \Omega$, and assume that every connected component of $\partial \Omega$ is a round circle.

Let $\Gamma(\Omega)$ denote the Schottky group of $\Omega$, that is, the free group of Möbius and anti-Möbius transformations generated by the family of reflections across the boundary circles.

Question : Can the limit set $\widehat{\mathbb{C}} \setminus \bigcup_{T \in \Gamma(\Omega)} T(\Omega)$ have positive area.

Remark : If there are only finitely many boundary circles, then the limit set is a Cantor set and it is not difficult to prove that it must have zero area. We can therefore assume that there are infinitely many boundary circles.

Thank you

Dear Mathoverflow Community,

Suppose that $\Omega$ is a domain in the Riemann Sphere $\widehat{\mathbb{C}}$ with $\infty \in \Omega$, and assume that every connected component of $\partial \Omega$ is a round circle.

Let $\Gamma(\Omega)$ denote the Schottky group of $\Omega$, that is, the free group of Möbius and anti-Möbius transformations generated by the family of reflections across the boundary circles.

Question : Can the limit set $\widehat{\mathbb{C}} \setminus \bigcup_{T \in \Gamma(\Omega)} T(\Omega)$ have positive area.

Remark : If there are only finitely many boundary circles, then the limit set is a Cantor set and it is not difficult to prove that it must have zero area. We can therefore assume that there are infinitely many boundary circles.

Thank you

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Malik Younsi
Malik Younsi
  • 2.2k
  • 2
  • 25
  • 44

Dear Mathoverflow Community,

I apologize in advance if the answer to the following question is well-known to experts :

Suppose that $\Omega$ is a domain in the Riemann Sphere $\widehat{\mathbb{C}}$ with $\infty \in \Omega$, and assume that every connected component of $\partial \Omega$ is a round circle.

Let $\Gamma(\Omega)$ denote the Schottky group of $\Omega$, that is, the free group of Möbius and anti-Möbius transformations generated by the family of reflections across the boundary circles.

Question : Can the limit set $\widehat{\mathbb{C}} \setminus \bigcup_{T \in \Gamma(\Omega)} T(\Omega)$ have zeropositive area?.

Remark : If there are only finitely many boundary circles, then the limit set is a Cantor set and it is not difficult to prove that it must have zero area. We can therefore assume that there are infinitely many boundary circles.

Thank you

Dear Mathoverflow Community,

I apologize in advance if the answer to the following question is well-known to experts :

Suppose that $\Omega$ is a domain in the Riemann Sphere $\widehat{\mathbb{C}}$ with $\infty \in \Omega$, and assume that every connected component of $\partial \Omega$ is a round circle.

Let $\Gamma(\Omega)$ denote the Schottky group of $\Omega$, that is, the free group of Möbius and anti-Möbius transformations generated by the family of reflections across the boundary circles.

Question : Can the limit set $\widehat{\mathbb{C}} \setminus \bigcup_{T \in \Gamma(\Omega)} T(\Omega)$ have zero area?

Remark : If there are only finitely many boundary circles, then the limit set is a Cantor set and it is not difficult to prove that it must have zero area. We can therefore assume that there are infinitely many boundary circles.

Thank you

Dear Mathoverflow Community,

I apologize in advance if the answer to the following question is well-known to experts :

Suppose that $\Omega$ is a domain in the Riemann Sphere $\widehat{\mathbb{C}}$ with $\infty \in \Omega$, and assume that every connected component of $\partial \Omega$ is a round circle.

Let $\Gamma(\Omega)$ denote the Schottky group of $\Omega$, that is, the free group of Möbius and anti-Möbius transformations generated by the family of reflections across the boundary circles.

Question : Can the limit set $\widehat{\mathbb{C}} \setminus \bigcup_{T \in \Gamma(\Omega)} T(\Omega)$ have positive area.

Remark : If there are only finitely many boundary circles, then the limit set is a Cantor set and it is not difficult to prove that it must have zero area. We can therefore assume that there are infinitely many boundary circles.

Thank you

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Malik Younsi
Malik Younsi
  • 2.2k
  • 2
  • 25
  • 44