Math 2.0

Group theory and generalizations Definition

Random group

Builds on Free groupPresentation of a groupHyperbolic groupFinite set

Definition

Fix an integer m≥2m \ge 2 and a real number dd with 0≤d≤10 \le d \le 1, the density. Let FmF_m be the free group on a1,…,ama_1, \dots, a_m. For each integer ℓ≥1\ell \ge 1, let CℓC_\ell be the finite set of cyclically reduced words of length ℓ\ell in FmF_m. Let RR be a subset of CℓC_\ell with ⌊(2m−1)dℓ⌋\floor{(2m - 1)^{d \ell}} elements, chosen uniformly at random among all such subsets. A random group at density dd and length ℓ\ell is the group G=⟨a1,…,am∣R⟩.G = \langle a_1, \dots, a_m \mid R \rangle. A property holds for random groups at density dd if the probability that GG has it tends to 11 as ℓ→∞\ell \to \infty.

Remarks

This is Gromov's density model. The number of cyclically reduced words of length ℓ\ell grows like (2m−1)ℓ(2m - 1)^\ell, so the density dd measures the number of relators on a logarithmic scale. Gromov's theorem: for d<12d < \frac{1}{2}, random groups at density dd are infinite, torsion-free and hyperbolic; for d>12d > \frac{1}{2}, random groups at density dd are trivial or of order 22.

Example: at density d=0d = 0, the set RR has exactly one element, so the model produces a random one-relator group.

Non-example: a single fixed group, such as ℤ2=⟨a,b∣aba−1b−1⟩\Z^2 = \langle a, b \mid aba^{-1}b^{-1} \rangle, is not "random" in itself. The definition describes a sequence of probability distributions on presentations, and only properties that hold with probability tending to 11 are meaningful.

Other models fix the number of relators and let their length grow (the few-relator model), or use relators of length 33 (the triangular model).

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionFree group, concept, no evidence levelFree groupconceptHyperbolic group, concept, no evidence levelHyperbolic groupconceptPresentation of a group, concept, no evidence levelPresentation of a gro…conceptProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionEquinumerous sets, definition, no evidence levelEquinumerous setsdefinitionFinite set, definition, no evidence levelFinite setdefinitionRandom group, definition, no evidence levelRandom groupdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Free group concept
  3. Hyperbolic group concept
  4. Presentation of a group concept
  5. Proposition concept
  6. Set definition
  7. Implication definition
  8. Predicate definition
  9. Ordered pair definition
  10. Existential quantifier definition
  11. Universal quantifier definition
  12. Cartesian product definition
  13. Subset definition
  14. Uniqueness quantifier definition
  15. Binary relation definition
  16. Function definition
  17. Codomain definition
  18. Injective function definition
  19. Surjective function definition
  20. Bijective function definition
  21. Equinumerous sets definition
  22. Finite set definition
  23. Random group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Random_group
About this page
TagC2ATM03215
URIm2://C2ATM03215
Kinddefinition
WritingW0 stub
MSC20P05, 20F65, 20F67
DefinesRandom group
UsesFree group, Presentation of a group, Hyperbolic group, Finite set
Changed2 November 2026

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