Definition:Grötzsch Annulus

Definition
Let $R \in \R: R > 1 $.

The set:
 * $A := \left\{{z \in \C: \left \vert {z}\right \vert > 1 \text{ and } z \notin \left[{R \ . \ . \ \infty}\right)}\right \}$

is called a Grötzsch annulus (or also Grötzsch extremal domain).

The modulus of $A$ is denoted $M \left({R}\right)$.

Properties
By the Grötzsch Modulus Theorem, among all annuli that separate the unit circle from the points $R$ and $\infty$, the Grötzsch annulus has the greatest modulus.

The Grötzsch annulus is closely related to the Teichmüller annulus.