Schottky group

Remarks

Example: let $\lambda \in \C$ with $\abs{\lambda} > 1$, $A_1(z) = \lambda z$, $D_1 = \set{\abs{z} < 1}$ and $D_1' = \set{\abs{z} > \abs{\lambda}} \cup \set{\infty}$. Their closures are disjoint, and $A_1$ maps the circle $\abs{z} = 1$ onto $\abs{z} = \abs{\lambda}$ and the region $\abs{z} > 1$ onto $D_1'$. So $\langle A_1 \rangle$ is a Schottky group of genus $1$; the quotient of $\C \setminus \set{0}$ by it is a torus.

Non-example: the group generated by $z \mapsto z + 1$ is not a Schottky group. Its generator is parabolic, while every element other than the identity of a Schottky group is loxodromic.

A Schottky group of genus $g$ is a free group of rank $g$ on $A_1, \ldots, A_g$ and is a Kleinian group. The common exterior of the $2g$ discs is a fundamental domain for its action on the domain of discontinuity $\Omega$. The quotient $\Omega / G$ is a compact Riemann surface of genus $g$.