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 (1994) and named after Robert Riley by them. Some subtle errors in their paper were corrected by Komori & Series (1998).

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.

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, MR 1272421
• Komori, Yohei; Series, Caroline (1998), "The Riley slice revisited", The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, pp. 303–316, arXiv:math/9810194, doi:10.2140/gtm.1998.1.303, MR 1668296