Non-amenable finitely presented torsion-by-cyclic groups

By A. Ol’shanskii, Mark V. Sapir. 2001

138 citations recorded by the source corpus.

Abstract. – We construct a finitely presented non-amenable group without free non-cyclic subgroups thus providing a finitely presented counterexample to von Neumann’s problem. Our group is an extension of a group of finite exponent n ≫ 1 by a cyclic group, so it satisfies the identity [x,y]n = 1.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/def1bc6b45e80ec3baa221ef78a800d6834408d5

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In Computational Theory and Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Groups of piecewise projective homeomorphisms (2012, background): “Finitely presented examples were constructed another twenty years later by Olshanskĭı–Sapir [11].”
Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity (2005, unknown): “Other examples were later given in [1,72].”
Milnor's Problem on the Growth of Groups and its Consequences (2011, result): “It is important to mention that both problems of Day had negative solutions not only for the class of finitely generated groups but also for the class of finitely presented groups, as was established by the author and Olshanskii and Sapir respectively [101, 189].”
Combinatorial algebra : syntax and semantics (2014, unknown): “Note, though, that the group from [264] satisfies a nontrivial identity ([x, y]n = 1, n ≥ 10(70), odd) while the group in [204] is lawless by Theorem 5.”
A nonamenable finitely presented group of piecewise projective homeomorphisms (2016, unknown): “Amenable, nitely presented, free group, piecewise, projective, ompson’s group, torsion free.”
Peripheral fillings of relatively hyperbolic groups (2005, background): “It plays an important role in some constructions of groups with ‘exotic’ properties [24].”
GROUP ACTIONS ON 1-MANIFOLDS: A LIST OF VERY CONCRETE OPEN QUESTIONS (2017, background): “Although it is not the first example of a group with these properties (see [115]), it has several other properties, as being torsion-free and of type F∞ (a property which is much stronger”