Riley slice

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 (Keen Series) and named after Robert Riley by them. Some subtle errors in their paper were corrected by (Komori Series).

Definition

The Riley slice consists of the complex numbers ρ such that the two matrices

[math]\displaystyle{ \begin{pmatrix}1&1\\0&1\\ \end{pmatrix}, \begin{pmatrix}1&0\\ \rho&1\\ \end{pmatrix} }[/math]

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.

See also

• Bers slice

References

• Keen, Linda; Series, Caroline (1994), "The Riley slice of Schottky space", Proceedings of the London Mathematical Society, Third Series 69 (1): 72–90, doi:10.1112/plms/s3-69.1.72, ISSN 0024-6115 
• Komori, Yohei; Series, Caroline (1998), "The Riley slice revisited", The Epstein birthday schrift, Geom. Topol. Monogr., 1, Geom. Topol. Publ., Coventry, pp. 303–316, doi:10.2140/gtm.1998.1.303