Schottky group

Fundamental domain of a Schottky group generated by 3 loxodromic isometries

In mathematics , Schottky groups are certain Kleinian groups that were first investigated by Friedrich Schottky in 1877 .

construction

Classic Schottky groups

We consider the Riemann sphere with the effect of by broken-linear transformations . ${\displaystyle P^{1}\mathbb {C} =\mathbb {C} \cup \left\{\infty \right\}}$ ${\displaystyle SL(2,\mathbb {C} )}$

Take pairs of disjoint disks ${\displaystyle 2k}$

${\displaystyle C_{-1},C_{1},\ldots ,C_{-k},C_{k}}$

in . For there are pictures , each of the interior of bijective on the exterior of mapping. The subgroup created by is a (classic) Schottky group. ${\displaystyle \mathbb {C} \cup \left\{\infty \right\}}$ ${\displaystyle i=1,\ldots ,k}$ ${\displaystyle \theta _{i}\in SL(2,\mathbb {C} )}$ ${\displaystyle C_{-i}}$ ${\displaystyle C_{i}}$ ${\displaystyle \theta _{1},\ldots ,\theta _{k}}$ ${\displaystyle \Gamma \subset SL(2,\mathbb {C} )}$

General Schottky groups

More generally, one can consider disjoint areas bounded by Jordan curves . If there are images that each map the inside of bijective to the outside of , then the group created by them is referred to as a Schottky group. In the context of this more general definition, the groups defined in the previous section are then referred to as classical Schottky groups. ${\displaystyle 2k}$ ${\displaystyle \theta _{i}\in SL(2,\mathbb {C} )}$ ${\displaystyle C_{-i}}$ ${\displaystyle C_{i}}$ ${\displaystyle \theta _{i}}$

properties

One can show that all Schottky groups are free groups and discrete subsets of . The first property follows from Klein's combination theorem and the second from Poincaré's polyhedron theorem . ${\displaystyle SL(2,\mathbb {C} )}$

Each non-elementary Kleinian group has numerous subgroups that are Schottky groups. The reason for this is that sufficiently high powers of given loxodromic isometries have disjoint "isometric circles" and therefore create a Schottky group.

Schottky groups in hyperbolic space

Hyperbolic handle bodies

The Riemann number sphere is the edge at infinity of the 3-dimensional hyperbolic space , the circles border hemispheres and they correspond to loxodromic isometries, which map the outside from bijective to the inside from . The quotient space is then homeomorphic to the interior of a handle body . ${\displaystyle H^{3}}$ ${\displaystyle C_{i}}$ ${\displaystyle S_{i}}$ ${\displaystyle \theta _{i}\in SL(2,\mathbb {C} )}$ ${\displaystyle S_{-i}}$ ${\displaystyle S_{i}}$ ${\displaystyle \Gamma \backslash H^{3}}$

Limes amount and discontinuity area

The Limes set of a Schottky group is a Cantor set . The complement is the discontinuity area . ${\displaystyle \Lambda (\Gamma )}$ ${\displaystyle P^{1}\mathbb {C} \setminus \Lambda (\Gamma )}$ ${\displaystyle \Omega (\Gamma )}$

The quotient is a Riemann surface . The union is a handle body. ${\displaystyle \Gamma \backslash \Omega (\Gamma )}$ ${\displaystyle \Gamma \backslash (H^{3}\cup \Omega (\Gamma ))}$

Characterization of Schottky groups

According to a theorem of Maskit , the following properties are equivalent to a discrete subgroup : ${\displaystyle \Gamma \subset SL(2,\mathbb {C} )}$

• ${\displaystyle \Gamma }$ is a Schottky group.
• ${\displaystyle \Gamma \backslash H^{3}}$is the inside of a handle body .
• ${\displaystyle \Gamma }$is a free group and all are loxodromic . ${\displaystyle \gamma \in \Gamma }$

For discrete subgroups whose elements are loxodromic, Hou has shown that they are classical Schottky groups if and only if the Hausdorff dimension of their Limes set is less than 1. ${\displaystyle \Gamma \subset SL(2,\mathbb {C} )}$

Schottky uniformization

A Schottky uniformization of a Riemann surface is given by a Schottky group , so that it is biholomorphic to the given Riemann surface. According to one of Koebe's theorem , every Riemann surface has a Schottky uniformization. ${\displaystyle \Gamma }$ ${\displaystyle \Gamma \backslash \Omega (\Gamma )}$

For a Riemann surface, one can even find a Schottky uniformization for every family of homologically independent simple closed curves , so that the given curves each border a circular disk in the handle body . ${\displaystyle \gamma _{1},\ldots ,\gamma _{s}}$ ${\displaystyle \gamma _{i}}$ ${\displaystyle \Gamma \backslash (H^{3}\cup \Omega (\Gamma ))}$

literature

• David Mumford , Caroline Series , David Wright : Indra's pearls. The vision of Felix Klein. Cambridge University Press, New York, 2002. ISBN 0-521-35253-3 .

Web links

Caroline Series: A crash course on Kleinian groups. Rend. Istit. Mat. Univ. Trieste 37 (2005), no. 1-2, 1-38 (2006). (Pp. 16–17)

Individual evidence

1. ^ Theorem 2.9 in: Matsuzaki-Taniguchi: Hyperbolic manifolds and Kleinian groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998. ISBN 0-19-850062-9
2. ↑ Theorem 4.23 in: Matsuzaki-Taniguchi, op.cit.
3. ^ Yong Hou: The classification of Kleinian groups of Hausdorff dimensions at most one