Combination set from Maskit

The combined set of Maskit is a theorem from the mathematical field of Klein groups .

It gives conditions for the constructibility of discrete groups of hyperbolic isometries as amalgamated products or HNN extensions.

It generalizes Klein's combination theorem and is therefore sometimes referred to as the Klein-Maskit combination theorem.

First combination set

Let small groups be such that it is a quasi-fox group . Be the two connected components of the complement of the limit set ${\displaystyle G_{1},G_{2}}$ ${\displaystyle H:=G_{1}\cap G_{2}}$ ${\displaystyle \Omega _{1},\Omega _{2}}$

${\displaystyle \Lambda (H):=S_{\infty }^{2}\setminus \Lambda (H)=\Omega _{1}\cup \Omega _{2}}$

and be

${\displaystyle H(\Omega _{i})\subset \Omega _{i},i=1,2}$

but

${\displaystyle g(\Omega _{i})\cap \Omega _{i}=\emptyset \ \forall g\in G_{i}\setminus H.}$

Then the subgroup created by and is a discrete group and is isomorphic to the amalgamated product ${\displaystyle G_{1}}$ ${\displaystyle G_{2}}$

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

Second combination set

Let be a Klein group and be quasi-Fuchs groups, which stabilize two different connected components of the discontinuity area . Be so ${\displaystyle G_{0}}$ ${\displaystyle H_{1},H_{2}\subset G}$ ${\displaystyle \Omega _{1},\Omega _{2}}$ ${\displaystyle \Omega (G_{0})\subset S_{\infty }^{2}}$ ${\displaystyle A\in Isom(H^{3})}$

${\displaystyle AH_{1}A^{-1}=H_{2}}$

induced an isomorphism from to and ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$

${\displaystyle A(\Omega _{1})=S_{\infty }^{2}\setminus cl(\Omega _{2}).}$

Then the subgroup created by and is a discrete group and is isomorphic to the HNN extension ${\displaystyle G_{0}}$ ${\displaystyle A}$

${\displaystyle G=G_{0}*_{A}}$.

literature

• Bernard Maskit : Kleinian groups. Basic teaching of the mathematical sciences 287. Springer-Verlag, Berlin, 1988. ISBN 3-540-17746-9 (Chapter VII)
• Michael Kapovich : Hyperbolic manifolds and discrete groups. Reprint of the 2001 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2009. ISBN 978-0-8176-4912-8 (Chapter 4.18)