// Workers AI · dad joke modeWhat did the "Bers slice" say? Bear-ly edible.

Computer generated image of the bers embedding of a punctured torus’ 2-dimensional Teichmüller space.

In the mathematical theory of Kleinian groups, Bers slices and Maskit slices, named after Lipman Bers and Bernard Maskit, are certain slices through the moduli space of Kleinian groups.

Bers slices

For a quasi-Fuchsian group, the limit set is a Jordan curve whose complement has two components. The quotient of each of these components by the groups is a Riemann surface, so we get a map from marked quasi-Fuchsian groups to pairs of Riemann surfaces, and hence to a product of two copies of Teichmüller space. A Bers slice is a subset of the moduli space of quasi-Fuchsian groups for which one of the two components of this map is a constant function to a single point in its copy of Teichmüller space.

The Bers slice gives an embedding of Teichmüller space into the moduli space of quasi-Fuchsian groups, called the Bers embedding, and the closure of its image is a compactification of Teichmüller space called the Bers compactification.

Maskit slices

A Maskit slice is similar to a Bers slice, except that the group is no longer quasi-Fuchsian, and instead of fixing a point in Teichmüller space one fixes a point in the boundary of Teichmüller space.

The Maskit boundary is a fractal in the Maskit slice separating discrete groups from more chaotic groups.

References

Gardiner, Frederick P. (1987), Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), New York: John Wiley & Sons, ISBN 978-0-471-84539-3, MR 0903027
Maskit, Bernard (1988), Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Berlin, New York: Springer-Verlag, ISBN 978-3-540-17746-3, MR 0959135
Minsky, Yair N. (1999), "The classification of punctured-torus groups", Annals of Mathematics, Second Series, 149 (2): 559–626, arXiv:math/9807001, doi:10.2307/120976, ISSN 0003-486X, MR 1689341
Mumford, David; Series, Caroline; Wright, David (2002), Indra's pearls, Cambridge University Press, ISBN 978-0-521-35253-6, MR 1913879

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