Schottky group
Encyclopedia
In mathematics, a Schottky group is a special sort of Kleinian group
, first studied by .
. Each Jordan curve not passing through p
divides the Riemann sphere into two pieces, and we call the piece containing p the "exterior" of the curve, and the other piece its "interior".
Suppose there are 2g disjoint Jordan curves A1, B1,..., Ag, Bg in the Riemann sphere
with disjoint interiors.
If there are Moebius transformations Ti taking the outside of Ai onto the inside of Bi, then the group generated by these transformations is a Kleinian group
. A Schottky group is any Kleinian group that can be constructed like this.
free group
s such that all non-trivial elements are loxodromic. Conversely showed that any finitely generated free Kleininan group such that all non-trivial elements are loxodromic is a Schottky group.
A fundamental domain for the action of a Schottky group G on its regular points Ω(G) in the Riemann sphere is given by the exterior of the Jordan curves defining it. The corresponding quotient space Ω(G)/G is given by joining up the Jordan curves in pairs, so is a compact Riemann surface of genus g. This is the boundary of the 3-manifold given by taking the quotient (H∪Ω(G))/G of 3-dimensional hyperbolic H space plus the regular set Ω(G) by the Schottky group G, which is a handlebody of genus g. Conversely any compact Riemann surface of genus g can be obtained from some Schottky group of genus g.
Schottky space of genus g is not simply connected in general, but its universal covering space can be identified with Teichmüller space
of compact genus g Riemann surfaces.
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
, first studied by .
Definition
Fix some point p on the Riemann sphereRiemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
. Each Jordan curve not passing through p
divides the Riemann sphere into two pieces, and we call the piece containing p the "exterior" of the curve, and the other piece its "interior".
Suppose there are 2g disjoint Jordan curves A1, B1,..., Ag, Bg in the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
with disjoint interiors.
If there are Moebius transformations Ti taking the outside of Ai onto the inside of Bi, then the group generated by these transformations is a Kleinian group
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
. A Schottky group is any Kleinian group that can be constructed like this.
Properties
Schottky groups are finitely generatedGenerating set of a group
In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
s such that all non-trivial elements are loxodromic. Conversely showed that any finitely generated free Kleininan group such that all non-trivial elements are loxodromic is a Schottky group.
A fundamental domain for the action of a Schottky group G on its regular points Ω(G) in the Riemann sphere is given by the exterior of the Jordan curves defining it. The corresponding quotient space Ω(G)/G is given by joining up the Jordan curves in pairs, so is a compact Riemann surface of genus g. This is the boundary of the 3-manifold given by taking the quotient (H∪Ω(G))/G of 3-dimensional hyperbolic H space plus the regular set Ω(G) by the Schottky group G, which is a handlebody of genus g. Conversely any compact Riemann surface of genus g can be obtained from some Schottky group of genus g.
Classical and non-classical Schottky groups
A Schottky group is called classical if all the disjoint Jordan curves corresponding to some set of generators can be chosen to be circles. gave an indirect and non-constructive proof of the existence of non-classical Schottky groups, and gave an explicit example of one. It has been shown by Peter that all finitely generated classical Schottky groups have limit sets of Hausdorff dimension bounded above strictly by a universal constant less than 2.Schottky space
Schottky space (of some genus g≥2) is the space of marked Schottky groups of genus g, in other words the space of sets of g elements of PSL2(C) that generate a Schottky group, up to equivalence under Moebius transformations . It is a complex manifold of complex dimension 3g−3. It contains classical Schottky space as the subset corresponding to classical Schottky groups.Schottky space of genus g is not simply connected in general, but its universal covering space can be identified with Teichmüller space
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
of compact genus g Riemann surfaces.