Finiteness Conditions on Groups

Q: What distinguishes FP2 from finite presentability?

FP2 is an algebraic condition on the relation module; finite presentability requires finitely many relations as a group. Bestvina–Brady constructed groups satisfying the first but not the second, settling a long-standing question.

Executive summary

Homological finiteness constrains group structure

The cohomological dimension of a group is the length of the shortest projective resolution of ℤ over its group ring. It is zero only for the trivial group, one exactly for free groups — the Stallings–Swan theorem — and infinite for any group with torsion. Finer finiteness conditions, FP_n and F_n, ask that resolutions be finitely generated or that classifying spaces have finite skeleta, and they distinguish groups that dimension alone cannot.

Learning objectives

Define cohomological and geometric dimension.
State the Stallings–Swan theorem.
Distinguish the FP_n and F_n conditions.
Explain why torsion forces infinite dimension.

Section 01Cohomological dimension

cd(G) = proj dim_ℤ[G] ℤ = sup { n : Hⁿ(G, M) ≠ 0 for some M }

Cohomological dimension of standard groups
Group	cd	Note
Trivial group	0	The only group with cd = 0
Free group	1	Stallings–Swan: cd = 1 characterises free groups
ℤⁿ	n	Classifying space is the n-torus
Surface group, genus ≥ 1	2	Poincaré duality group
Finite non-trivial group	∞	Torsion forces periodic non-vanishing
SL_n(ℤ)	∞	Contains torsion; torsion-free subgroups have finite cd

Torsion is the obstruction

A finite cyclic subgroup has periodic non-vanishing cohomology, and restriction is injective on the relevant primary part, so cohomology of the ambient group cannot vanish in high degrees. Hence any group with torsion has infinite cohomological dimension — no exceptions.

Section 02Stallings–Swan and geometric dimension

The geometric dimension gd(G) is the least dimension of a classifying space. Always cd ≤ gd, with equality except possibly when cd = 2 and gd = 3 — the Eilenberg–Ganea problem, still open.

AlgorithmStallings–Swanin: a group of cohomological dimension at most 1 → out: freeness

Suppose cd(G) ≤ 1, so ℤ has a projective resolution of length 1 over ℤ[G].
Stallings proved the finitely generated case using ends of groups.
Swan extended it to arbitrary groups.
Conclusion: G is free. The converse is immediate, since a free group has a one-dimensional classifying space — a wedge of circles.
So cd(G) ≤ 1 ⇔ G free, and freeness becomes a homological condition.

Section 03Finiteness properties

FPnAlgebraic

ℤ admits a projective resolution over ℤ[G] that is finitely generated in degrees up to n. FP₁ is equivalent to finite generation of G.

FnGeometric

G has a classifying space with finite n-skeleton. F₁ is finite generation; F₂ is finite presentability.

F_n implies FP_n, and the converse holds given finite presentability. Bestvina–Brady groups separate the conditions: there are groups that are FP₂ but not finitely presented.

Class

Duality groups

Groups whose cohomology satisfies a duality with homology in complementary degree, generalising Poincaré duality. Surface groups and torsion-free arithmetic groups are examples.

Class

Groups of type FP_∞

Finitely generated in every degree. Includes many groups arising geometrically, and the condition behaves well under extensions.

Class

Virtual duality groups

Have a finite-index subgroup that is a duality group. SL_n(ℤ) is one, which is how its cohomology is studied despite having torsion.

ReferenceFrequently asked questions

Is the Eilenberg–Ganea problem still open?

Yes. It asks whether a group with cd = 2 must have gd = 2. No counterexample is known and no proof exists; it is one of the long-standing open problems in the homological theory of groups.

Why does virtual cohomological dimension help?

Because a group with torsion has infinite cd, but often has a torsion-free finite-index subgroup with finite cd. That number — the virtual cohomological dimension — is well defined by Serre's theorem and carries the useful information.

What distinguishes FP2 from finite presentability?

FP₂ is an algebraic condition on the relation module; finite presentability requires finitely many relations as a group. Bestvina–Brady constructed groups satisfying the first but not the second, settling a long-standing question.

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ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Finiteness Conditions on Groups. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat Finiteness Conditions on Groups as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—groups, finiteness, dimension, conditions, cohomological—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Finiteness Conditions on Groups?