Ping pong lemma

In the mathematical field of group theory , the ping-pong lemma is a method for constructing free subgroups of a group. It is attributed to Felix Klein , who used it in the 1870s as "the process of nesting" in the study of Kleinian groups . The formulation given below goes back to Jacques Tits , who used it in the early 1970s (as "a criterion of freedom") to prove the Tits alternative .

Ping pong lemma

A group works in one room . Let be nontrivial subgroups with at least three elements and let there be disjoint subsets such that for all ${\displaystyle G}$ ${\displaystyle X}$ ${\displaystyle H_{1},\ldots ,H_{n}\subset G}$ ${\displaystyle X_{1},\ldots ,X_{n}\subset X}$

${\displaystyle h\in H_{i}\setminus \left\{1\right\}}$

and inclusion for everyone ${\displaystyle j\not =i}$

${\displaystyle h(X_{j})\subset X_{i}}$

applies. Then is a free product . ${\displaystyle G}$ ${\displaystyle G=H_{1}*\ldots *H_{n}}$

example

The ones from the matrices

${\displaystyle A={\begin{pmatrix}1&2\\0&1\end{pmatrix}}}$ and ${\displaystyle B={\begin{pmatrix}1&0\\2&1\end{pmatrix}}}$

The created subgroup is a free group. ${\displaystyle G\subset SL(2,\mathbb {Z} )}$

To prove this, consider the linear action and apply the ping-pong lemma to the subsets ${\displaystyle \mathbb {R} ^{2}}$

${\displaystyle X_{1}=\left\{{\begin{pmatrix}x\\y\end{pmatrix}}\in \mathbb {R} ^{2}:|x|>|y|\right\}}$

${\displaystyle X_{2}=\left\{{\begin{pmatrix}x\\y\end{pmatrix}}\in \mathbb {R} ^{2}:|x|<|y|\right\}.}$

on.

More generally, the ping-pong lemma is used to prove the lemma of Sanov : If there are complex numbers with , then generate ${\displaystyle z_{1},z_{2}}$ ${\displaystyle |z_{1}z_{2}|\geq 4}$

${\displaystyle A={\begin{pmatrix}1&z_{1}\\0&1\end{pmatrix}}}$ and ${\displaystyle B={\begin{pmatrix}1&0\\z_{2}&1\end{pmatrix}}}$

a free subset of . ${\displaystyle SL(2,\mathbb {C} )}$

Applications

• In the theory of Klein's groups one can use the ping-pong lemma for the construction of Schottky groups : one has pairwise disjoint circular disks in and for there are mappings , each of which map the interior of bijective to the exterior of . Then the subgroup created by is a free group called a Schottky group. It can be shown that every non-elementary Klein group contains a Schottky group of rank . ${\displaystyle 2k}$ ${\displaystyle C_{-1},C_{1},\ldots ,C_{-k},C_{k}}$ ${\displaystyle \mathbb {C} \cup \left\{\infty \right\}}$ ${\displaystyle i=1,\ldots ,k}$ ${\displaystyle \theta _{i}\in SL(2,\mathbb {C} )}$ ${\displaystyle C_{-i}}$ ${\displaystyle C_{i}}$ ${\displaystyle \theta _{1},\ldots ,\theta _{k}}$ ${\displaystyle \Gamma \subset SL(2,\mathbb {C} )}$ ${\displaystyle \geq 2}$
• The ping-pong lemma was used in proving the tits alternative . In its classic form, this said that a finitely generated and not almost resolvable subgroup of contains a free subgroup; it can now also be proven more generally for finitely generated and not almost resolvable subgroups, for example of hyperbolic groups , mapping class groups and automorphism groups of free groups. ${\displaystyle GL(n,K)}$

literature

Pierre de la Harpe. Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago. ISBN 0-226-31719-6 ; Ch. II.B "The table tennis lemma (Klein's criterion) and examples of free products"

Web links

• Chapter 4.4 in Geometric group theory, an introduction

Individual evidence

1. ^ Proposition 1.1 in: Tits, J .: "Free subgroups in linear groups". Journal of Algebra 20 (2), 250-270 (1972). online (pdf)