Lukas Boeke - Actions on R and orderings I

Abstract

Introduce orderability and give some (non-)examples. Prove Theorem 6.8. State Holder’s Theorem and describe steps of the proof. State and prove Solodov’s Theorem 6.12. Define convergence groups and prove the easy direction of Theorem 6.14.

Talk

Definition

A group is called left-orderable if there exists a total order, which is invariant under left-multiplication. Vgl. Ordnungserhaltender Zopf

Example

$\mathbb{Z}$ has an obvious order. ${\mathbb{Z}}^2$ has the lexicographical order. Every product of orderable groups is orderable

Theorem

Let $\Gamma$ be an orderable group. Then the following are equivalent

1. $\Gamma$ acts faithfully on $\mathbb{R}$ by order-preserving homeos
2. $\gamma$ is left-orderable

Proof: For 1 implies 2: We seperate the proof in two cases. In the first case $\Gamma$ acts freely on $\mathbb{R}$. Two elements ${\gamma }_1,{\gamma }_2\in \Gamma$ have ${\gamma }_1\le {\gamma }_2$ iff ${\gamma }_1{x}_0\le {\gamma }_2{x}_0$. If it does not act freely, we do some funny stuff with dense sequences. 2 implies 1 if the group is countable. Then we can give an algorithm that constructs a map $\nu :\Gamma \to \mathbb{R}$. The order of $\nu (\Gamma )$ is preserved by the action induced by left-multiplication. This can be extended to $\mathbb{R}$.

Assume $\Gamma$ has a bi-ordering. then for all $\gamma \in \Gamma \backslash \{e\}$ and for all $x\in \mathbb{R}$ we have either $\gamma x\ge x$ or $\gamma x\le x$.

Theorem (Hölder)

If $\Gamma$ acts on the reals by homeos then $\Gamma$ must be abelian. It must be a subset of $\mathbb{R}$ and semi-conjugate to a group of translations.