Combination theorem from Klein

The combination set of small is a theorem from the mathematical field of fox's and Klein groups .

It gives conditions for the constructibility of discrete groups of hyperbolic isometries as free products and is used, for example, in the construction of Schottky groups .

It was proven by Felix Klein in 1883 . Occasionally the more general Maskit combination set is also referred to as the Klein-Maskit combination set .

Combination set

The combination set has two formulations, one for isometrics of the hyperbolic space and an equivalent one for Möbius transformations. (The latter was originally proven by Felix Klein.) The equivalence of the two statements is obtained from the fact that isometrics of the 3-dimensional hyperbolic space act as Möbius transformations on the "sphere in infinity" . ${\displaystyle H^{3}}$ ${\displaystyle S_{\infty }^{2}=\partial _{\infty }H^{3}}$

Combination set for discrete groups of hyperbolic isometries

Let be two discrete subgroups of (the group of the orientation- preserving isometries of hyperbolic space ) with fundamental domains that define the conditions ${\displaystyle G_{1},G_{2}}$ ${\displaystyle Isom^{+}(H^{n})}$ ${\displaystyle D_{1},D_{2}\subset H^{n}}$

${\displaystyle int(D_{1})\supset ext(D_{2}),int(D_{2})\supset ext(D_{1})}$

fulfill. Then the subgroup created by and is a discrete subgroup and isomorphic to the free product ${\displaystyle G_{1}}$ ${\displaystyle G_{2}}$ ${\displaystyle G\subset Isom(H^{n})}$

${\displaystyle G=G_{1}*G_{2}.}$

A fundamental domain for the effect of on is ${\displaystyle G}$ ${\displaystyle H^{n}}$

${\displaystyle D=D_{1}\cap D_{2}}$.

Combination set for furniture transformations

Let be two discrete groups of Möbius transformations , i.e. discrete subgroups of with fundamental domains that the conditions ${\displaystyle G_{1},G_{2}}$ ${\displaystyle PSL(2,\mathbb {C} )}$ ${\displaystyle D_{1},D_{2}\subset S_{\infty }^{2}}$

${\displaystyle int(D_{1})\supset ext(D_{2}),int(D_{2})\supset ext(D_{1})}$

fulfill. Then the subgroup created by and is a discrete subgroup and isomorphic to the free product ${\displaystyle G_{1}}$ ${\displaystyle G_{2}}$ ${\displaystyle G\subset PSL(2,\mathbb {C} )}$

${\displaystyle G=G_{1}*G_{2}.}$

A fundamental domain for the effect of on is ${\displaystyle D}$ ${\displaystyle S_{\infty }^{2}}$

${\displaystyle D=D_{1}\cap D_{2}}$.

Proof idea

For every point and every reduced word ${\displaystyle x\in D}$

${\displaystyle g=g_{1}g_{2}\ldots g_{n-1}g_{n}}$

with shows by complete induction . For a detailed proof of a more general statement see:. ${\displaystyle g_{2k-1}\in G_{1},g_{2k}\in G_{2}}$ ${\displaystyle gx\not \in D}$

Application: Schottky groups

→ Main article : Schottky group

Construction of Schottky groups : Let Möbius transformations and Jordan curves , so that the inside of is mapped to the outside of in each case . Then the group created by is discrete and a free group . (Groups constructed in this way are called Schottky groups.) ${\displaystyle g_{1},\ldots ,g_{k}\in PSL(2,\mathbb {C} )}$ ${\displaystyle A_{1},B_{1},\ldots ,A_{k},B_{k}\subset S_{\infty }^{2}}$ ${\displaystyle i=1,\ldots ,k}$ ${\displaystyle g_{i}}$ ${\displaystyle A_{i}}$ ${\displaystyle B_{i}}$ ${\displaystyle g_{1},\ldots ,g_{k}}$

The above construction can also be derived from Poincaré's polyhedron without the general combination theorem. With the combination theorem one can prove the following stronger statement: Every non-elementary (not necessarily discrete) group contains a non-elementary Schottky group.

Schottky groups are convex-cocompact . Your Limes set is a Cantor set .

literature

• Felix Klein : New contributions to the Riemannian function theory , Math. Ann. 21: 141-218 (1883).
• R. Fricke and F. Klein: Lectures on the theory of automorphic functions. I , Teubner, Leipzig, 1897.
• LR Ford: Automorphic functions , 1st ed., McGraw-Hill, New York, 1929.
• Bernard Maskit : Kleinian groups. Basic teaching of mathematical sciences 287. Springer-Verlag, Berlin, 1988. ISBN 3-540-17746-9

Individual evidence

1. ↑ For a closed subset we denote with the open kernel and with the complement of . ${\displaystyle D\subset S^{2}}$ ${\displaystyle int(D)}$ ${\displaystyle ext(D):=S^{2}\setminus D}$ ${\displaystyle D}$
2. ^ Maskit, On Klein's combination theorem , Trans. Amer. Math. Soc. 120: 499-509 (1965) online
3. ↑ Every Jordan curve is broken down into two related components ( Jordan's curve theorem ). We choose an (arbitrary) fixed point and then define the "inside" of a Jordan curve as the connected component that contains the outside as the complement. ${\displaystyle S_{\infty }^{2}}$ ${\displaystyle p\in S_{\infty }^{2}}$ ${\displaystyle p}$