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