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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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