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 kn ...
and
Bernard Maskit
Bernard Maskit is an American mathematician known for his expertise in Kleinian groups. The Maskit slice through the moduli space of Kleinian groups is named after him; he is the author of the book ''Kleinian Groups'' (Grundlehren der Mathematisc ...
, are certain slices through the
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
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. If the limit set is equal to the Jordan curve the quasi-Fuchsian group is said to be of type one, a ...
, 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. For example, the function is a constant function because the value of is 4 regardless of the input value (see image).
Basic properties ...
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
Compactification may refer to:
* Compactification (mathematics), making a topological space compact
* Compactification (physics), the "curling up" of extra dimensions in string theory
See also
* Compaction (disambiguation)
Compaction may refer t ...
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 is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
, 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