Riley slice

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title: "Riley slice" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["kleinian-groups"] topic_path: "general/kleinian-groups" source: "https://en.wikipedia.org/wiki/Riley_slice" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In the mathematical theory of Kleinian groups, the Riley slice of Schottky space is a family of Kleinian groups generated by two parabolic elements. It was studied in detail by and named after Robert Riley by them. Some subtle errors in their paper were corrected by .

Definition

The Riley slice consists of the complex numbers ρ such that the two matrices : \begin{pmatrix}1&1\0&1\ \end{pmatrix}, \begin{pmatrix}1&0\ \rho&1\ \end{pmatrix} generate a Kleinian group G with regular set Ω such that Ω/G is a 4-times punctured sphere.

The Riley slice is the quotient of the Teichmuller space of a 4-times punctured sphere by a group generated by Dehn twists around a curve, and so is topologically an annulus.

References

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