# Finiteness properties of groups

In mathematics, **finiteness properties** of a group are a collection of properties that allow the use of various algebraic and topological tools, for example group cohomology, to study the group. It is mostly of interest for the study of infinite groups.

Special cases of groups with finiteness properties are finitely generated and finitely presented groups.

## Topological finiteness properties

Given an integer *n* ≥ 1, a group is said to be *of type* *F*_{n} if there exists an aspherical CW-complex whose fundamental group is isomorphic to (a classifying space for ) and whose *n*-skeleton is finite. A group is said to be of type *F*_{∞} if it is of type *F*_{n} for every *n*. It is of type *F* if there exists a finite aspherical CW-complex of which it is the fundamental group.

For small values of *n* these conditions have more classical interpretations:

- a group is of type
*F*_{1}if and only if it is finitely generated (the Cayley graph is the finite 1-skeleton of a classifying space);

- a group is of type
*F*_{2}if and only if it is finitely presented (using the Cayley complex instead of the Cayley graph).

It is known that for every *n* ≥ 1 there are groups of type *F*_{n} which are not of type *F*_{n+1}. Finite groups are of type *F*_{∞} but not of type *F*.

A reformulation of the *F*_{n} property is that a group has it if and only if it acts properly discontinuously, freely and cocompactly on a CW-complex whose homotopy groups vanish. Another finiteness property can be formulated by replacing homotopy with homology: a group is said to be of type *FH*_{n} if it acts as above on a CW-complex whose *n* first homology groups vanish.

## Algebraic finiteness properties

Let be a group and its group ring. The group is said to be of type FP_{n} if there exists a resolution of the trivial -module such that the *n* first terms are finitely generated projective -modules.[1] The types *FP*_{∞} and *FP* are defined in the obvious way.

The same statement with projective modules replaced by free modules defines the classes *FL*_{n} for *n* ≥ 1, *FL*_{∞} and *FL*.

It is also possible to define classes *FP*_{n}(*R*) and *FL*_{n}(*R*) for any commutative ring *R*, by replacing the group ring by in the definitions above.

Either of the conditions *F*_{n} or *FH*_{n} imply *FP*_{n} and *FL*_{n} (over any commutative ring). A group is of type *FP*_{1} if and only if it is finitely generated,[1] but for any *n* ≥ 2 there exists groups which are of type *FP*_{n} but not *F*_{n}.[2]

## Group cohomology

If a group is of type *FP*_{n} then its cohomology groups are finitely generated for . If it is of type *FP* then it is of finite cohomological dimension. Thus finiteness properties play an important role in the cohomology theory of groups.

## Examples

### Finite groups

A finite group acts freely on the unit sphere in , preserving a CW-complex structure with finitely many cells in each dimension.[3] Since this unit sphere is contractible, every finite group is of type F_{∞}.

A non-trivial finite group is never of type *F* because it has infinite cohomological dimension. This also implies that a group with a non-trivial torsion subgroup is never of type *F*.

### Nilpotent groups

If is a torsion-free, finitely generated nilpotent group then it is of type F.[4]

### Geometric conditions for finiteness properties

Negatively curved groups (hyperbolic or CAT(0) groups) are always of type *F*_{∞}.[5][6] Such a group is of type *F* if and only if it is torsion-free.

As an example, cocompact S-arithmetic groups in algebraic groups over number fields are of type F_{∞}. The Borel–Serre compactification shows that this is also the case for non-cocompact arithmetic groups.

Arithmetic groups over function fields have very different finiteness properties: if is an arithmetic group in a simple algebraic group of rank over a global function field (such as ) then it is of type F_{r} but not of type F_{r+1}.[7]

## Notes

- Brown 1982, p. 197.
- Bestvina, Mladen; Brady, Noel (1997), "Morse theory and finiteness properties of groups",
*Inventiones Mathematicae*,**129**(3): 445–470, Bibcode:1997InMat.129..445B, doi:10.1007/s002220050168 - Brown 1982, p. 20.
- Brown 1982, p. 213.
- Bridson 1999, p. 439.
- Bridson 1999, p. 468.
- Bux, Kai-Uwe; Köhl, Ralf; Witzel, Stefan (2013). "Higher finiteness properties of reductive arithmetic groups in positive characteristic: The Rank Theorem".
*Annals of Mathematics*.**177**: 311–366. arXiv:1102.0428. doi:10.4007/annals.2013.177.1.6.