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Groups of piecewise linear homeomorphisms of the real line. (English) Zbl 0563.57022
The paper studies the group PLF($${\mathbb{R}})$$ of orientation-preserving homeomorphisms of the real line $${\mathbb{R}}$$ which are piecewise-linear with respect to a finite subdivision of $${\mathbb{R}}$$. The main results are presentations of PLF($${\mathbb{R}})$$ and certain of its subgroups; a proof that PLF($${\mathbb{R}})$$ contains no free subgroups of rank greater than 1; and a proof that PLF($${\mathbb{R}})$$ satisfies no laws. Also constructed is a sequence of finitely-presented subgroups G(p) of PLF($${\mathbb{R}})$$ which contain no free subgroups of rank greater than 1 and satisfy no laws. (R. Geoghegan has suggested the possibility that among the G(p)’s there might be a finitely presented counterexample to a conjecture of von Neumann: A discrete group is non-amenable if and only if it contains a subgroup isomorphic to the free group of rank 2.)

##### MSC:
 57S25 Groups acting on specific manifolds 20F05 Generators, relations, and presentations of groups 20F60 Ordered groups (group-theoretic aspects) 57Q99 PL-topology
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