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

$\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right),\left(\begin{array}{cc}1& 0\\ \rho & 1\end{array}\right)$

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