mathematics

Von Neumann conjecture

Von Neumann conjecture is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Von Neumann conjecture rather than just read about it. In short: In mathematics, the von Neumann conjecture stated that a group G is non-amenable if and only if G contains a subgroup that is a free group on two generators. The conjecture was disproved in 1980.

Key takeaways

Von Neumann conjecture belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Von Neumann conjecture to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Von Neumann conjecture from memory before moving on to harder problems.

Reference excerpt

In mathematics, the von Neumann conjecture stated that a group G is non-amenable if and only if G contains a subgroup that is a free group on two 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 groups and showed that no amenable group contains a free subgroup of rank 2. The suggestion that the 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 groups. The historically first potential counterexample is 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 groups, 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 groups are also counterexamples. None of these counterexamples are 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 homeomorphisms 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 Tatch Moore isolated a finitely presented non-amenable subgroup of Monod's group. This provides the first torsion-free finitely presented counterexample, and admits a presentation with 3 generators and 9 relations. Lodha later showed that this group satisfies the property F ∞ {\displaystyle F_{\infty }} , which is a stronger finiteness property.

References Adian, Sergei (1982), "Random walks on free periodic groups", Izv. Akad. Nauk SSSR, Ser. Mat. (in Russian), 46 (6): 1139–1149, 1343, Zbl 0512.60012 Day, Mahlon M. (1957), "Amenable semigroups", Ill. J. Math., 1: 509–544, Zbl 0078.29402 Ol'shanskii, Alexander (1980), "On the question of the existence of an invariant mean on a group", Uspekhi Mat. Nauk (in Russian), 35 (4): 199–200, Zbl 0452.20032 Ol'shanskii, Alexander; Sapir, Mark (2003), "Non-amenable finitely presented torsion-by-cyclic groups", Publications Mathématiques de l'IHÉS, 96 (1): 43–169, arXiv:math/0208237, doi:10.1007/s10240-002-0006-7, S2CID 122990460, Zbl 1050.20019 Monod, Nicolas (2013), "Groups of piecewise projective homeomorphisms", Proceedings of the National Academy of Sciences of the United States of America, 110 (12): 4524–4527, arXiv:1209.5229, Bibcode:2013PNAS..110.4524M, doi:10.1073/pnas.1218426110, Zbl 1305.57002 Lodha, Yash; Moore, Justin Tatch (2016), "A nonamenable finitely presented group of piecewise projective homeomorphisms", Groups, Geometry, and Dynamics, 10 (1): 177–200, arXiv:1308.4250v3, doi:10.4171/GGD/347, MR 3460335 Lodha, Yash (2020), "A nonamenable type F ∞ {\displaystyle F_{\infty }} group of piecewise projective homeomorphisms", Journal of Topology, 13 (4): 1767–1838, doi:10.1112/topo.12172, S2CID 228915338

Worked examples

Example 1 — a first encounter with Von Neumann conjecture

Start with the simplest possible case. Write down what Von Neumann conjecture claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Von Neumann conjecture before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Von Neumann conjecture ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Von Neumann conjecture

In research: Von Neumann conjecture appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Von Neumann conjecture in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Von Neumann conjecture is common in secondary-school and first-year university syllabi. It links to neighbouring topics Combinatorial group theory, Disproved conjectures, Geometric group theory, so understanding it makes those chapters shorter.
In everyday life: Look for Von Neumann conjecture outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Von Neumann conjecture in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Von Neumann conjecture means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Von Neumann conjecture out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Von Neumann conjecture in simple terms?

In mathematics, the von Neumann conjecture stated that a group G is non-amenable if and only if G contains a subgroup that is a free group on two generators. The conjecture was disproved in 1980.

Why does Von Neumann conjecture matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Von Neumann conjecture?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Von Neumann conjecture.