Editing Von Neumann conjecture

{{Short description|Disproven mathematical theory concerning Banach-Tarski and amenable groups}}
In [[mathematics]], the '''von Neumann conjecture''' stated that a [[group (mathematics)|group]] ''G'' is non-[[amenable group|amenable]] [[if and only if]] ''G'' contains a [[subgroup]] that is a [[free group]] on two [[generating set of a group|generators]]. The [[conjecture]] was disproved in 1980.

In 1929, during his work on the [[Banach–Tarski paradox]], [[John von Neumann]] defined the concept of [[amenable group]]s and showed that no amenable group contains a free subgroup of [[free group|rank]] 2. The suggestion that the [[converse (logic)|converse]] might hold, that is, that every non-amenable group contains a free subgroup on two generators, was made by a number of different authors in the 1950s and 1960s. Although von Neumann's name is popularly attached to the conjecture, its first written appearance seems to be due to [[Mahlon Marsh Day]] in 1957.

The [[Tits alternative]] is a fundamental [[theorem]] which, in particular, establishes the conjecture within the class of [[linear group]]s.

The historically first potential [[counterexample]] is [[Thompson groups|Thompson group ''F'']]. While its amenability is a wide-[[open problem]], the general conjecture was shown to be false in 1980 by [[Alexander Ol'shanskii]]; he demonstrated that [[Tarski monster group]]s, constructed by him, which are easily seen not to have free subgroups of rank 2, are not amenable. Two years later, [[Sergei Adian]] showed that certain [[Burnside group]]s are also counterexamples. None of these counterexamples are [[finitely presented group|finitely presented]], and for some years it was considered possible that the conjecture held for finitely presented groups. However, in 2003, Alexander Ol'shanskii and [[Mark Sapir]] exhibited a collection of finitely presented groups which do not satisfy the conjecture.

In 2013, [[Nicolas Monod]] found an easy counterexample to the conjecture. Given by piecewise projective [[homeomorphism]]s of the line, the group is remarkably simple to understand. Even though it is not amenable, it shares many known properties of amenable groups in a straightforward way. In 2013, Yash Lodha and [[Justin T. Moore|Justin Tatch Moore]] isolated a finitely presented non-amenable subgroup of Monod's group. This provides the first [[torsion-free group|torsion-free]] finitely presented counterexample, and admits a [[presentation of a group|presentation]] with 3 generators and 9 relations. Lodha later showed that this group satisfies the [[Finiteness properties of groups|property <math>F_{\infty}</math>]], which is a stronger finiteness property.

==References==
*{{Citation |last=Adian |first=Sergei |authorlink=Sergei Adian |title=Random walks on free periodic groups |journal=Izv. Akad. Nauk SSSR, Ser. Mat. |volume=46 |year=1982 |issue=6 |pages=1139–1149, 1343 |language=Russian | zbl=0512.60012 }}
*{{citation | last=Day | first=Mahlon M. | title=Amenable semigroups | zbl=0078.29402 | journal=Ill. J. Math. | volume=1 | pages=509–544 | year=1957 }}
*{{Citation |last=Ol'shanskii |first=Alexander |title=On the question of the existence of an invariant mean on a group |journal=Uspekhi Mat. Nauk |volume=35 |year=1980 |issue=4 |pages=199–200 |language=Russian | zbl=0452.20032 }}
*{{Citation |last1=Ol'shanskii |first1=Alexander |last2=Sapir |first2=Mark |title=Non-amenable finitely presented torsion-by-cyclic groups |journal=[[Publications Mathématiques de l'IHÉS]] |volume=96 |year=2003 |issue=1 |pages=43–169 |doi=10.1007/s10240-002-0006-7 | zbl=1050.20019 |arxiv=math/0208237 |s2cid=122990460 }}
*{{Citation |last=Monod |first=Nicolas |authorlink=Nicolas Monod |title=Groups of piecewise projective homeomorphisms |journal=[[Proceedings of the National Academy of Sciences of the United States of America]] |volume=110 |year=2013 |issue=12 |pages=4524–4527 |doi=10.1073/pnas.1218426110 | zbl=1305.57002 |arxiv=1209.5229 |bibcode=2013PNAS..110.4524M |doi-access=free }}
*{{Citation |last1=Lodha |first1=Yash|last2=Moore |first2=Justin Tatch |title=A nonamenable finitely presented group of piecewise projective homeomorphisms |journal=[[Groups, Geometry, and Dynamics]]|volume= 10| year=2016| issue= 1| pages= 177–200 |mr=3460335 | doi=10.4171/GGD/347 |arxiv=1308.4250v3 }}
*{{Citation |last=Lodha |first=Yash |title=A nonamenable type <math>F_{\infty}</math> group of piecewise projective homeomorphisms |journal=[[Journal of Topology]]|volume= 13| year=2020| issue= 4| pages= 1767–1838| doi=10.1112/topo.12172|s2cid=228915338 }}


{{DEFAULTSORT:Von Neumann Conjecture}}
[[Category:Topological groups]]
[[Category:Disproved conjectures]]
[[Category:Geometric group theory]]
[[Category:Combinatorial group theory]]