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Soluble right orderable groups are locally indicable. (English) Zbl 0802.20035

A right ordered group is a group \(G\) with a total ordering \(\leq\) such that for any \(x, y, z \in G\), \(x \leq y \Rightarrow xz \leq yz\). A group is said to be locally indicable if every non-trivial finitely generated subgroup admits a homomorphism onto the infinite cyclic group. We shall say that a group is RO-simple if it is non-trivial, right orderable but has no non-trivial proper right orderable quotients. The main results: Theorem A. Let \(G\) be a soluble-by-finite group. Then \(G\) is right orderable iff \(G\) is locally indicable. Theorem B. Let \(G\) be a finitely generated RO-simple group. Then (i) every abelian normal subgroup of \(G\) is central; (ii) no non-trivial element of the centre \(Z(G)\) of \(G\) is a commutator; (iii) \(G/Z(G)\) has no non-trivial abelian normal subgroups; and (iv) if \(G\) is non-abelian and has non-trivial centre then the second bounded cohomology group \(H^ 2_ b(G,\mathbb{R})\) is non-trivial and \(G\) is non-amenable. The proof identifies an interesting connection between the theory of right orderable groups and the theory of amenable groups and bounded cohomology.
Reviewer: F.Šik (Brno)

MSC:

20F60 Ordered groups (group-theoretic aspects)
20F16 Solvable groups, supersolvable groups
06F15 Ordered groups
20E25 Local properties of groups
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