In the mathematical theory of
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
s, Bers slices and Maskit slices, named after
Lipman Bers
Lipman Bers ( Latvian: ''Lipmans Berss''; May 22, 1914 – October 29, 1993) was a Latvian-American mathematician, born in Riga, who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups. He was also k ...
and
Bernard Maskit, are certain slices through the
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
of Kleinian groups.
Bers slices
For a
quasi-Fuchsian group In the mathematical theory of Kleinian groups, a quasi-Fuchsian group is a Kleinian group whose limit set is contained in an invariant Jordan curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a lin ...
, 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
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
, 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
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...
. 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
In mathematics, a constant function is a function whose (output) value is the same for every input value.
Basic properties
As a real-valued function of a real-valued argument, a constant function has the general form or just For example, ...
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
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*{{Citation , last1=Mumford , first1=David , author1-link=David Mumford , last2=Series , first2=Caroline , last3=Wright , first3=David , title=Indra's pearls , url=http://klein.math.okstate.edu/IndrasPearls/ , publisher=
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
, isbn=978-0-521-35253-6 , mr=1913879 , year=2002
External links
Pictures of Bers slicesMaskit sliceBers slice for square torusBers slice for hexagonal torus
Kleinian groups