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 $$\Gamma$$ is said to be of type Fn if there exists an aspherical CW-complex whose fundamental group is isomorphic to $$\Gamma$$ (a classifying space for $$\Gamma$$) and whose n-skeleton is finite. A group is said to be of type F∞ if it is of type Fn 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 F1 if and only if it is finitely generated (the rose with petals indexed by a finite generating family is the 1-skeleton of a classifying space, the Cayley graph of the group for this generating family is the 1-skeleton of its universal cover);


 * a group is of type F2 if and only if it is finitely presented (the presentation complex, i.e. the rose with petals indexed by a finite generating set and 2-cells corresponding to each relation, is the 2-skeleton of a classifying space, whose universal cover has the Cayley complex as its 2-skeleton).

It is known that for every n ≥ 1 there are groups of type Fn which are not of type Fn+1. Finite groups are of type F∞ but not of type F. Thompson's group $$F$$ is an example of a torsion-free group which is of type F∞ but not of type F.

A reformulation of the Fn 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 $$\pi_0, \ldots, \pi_{n-1}$$ vanish. Another finiteness property can be formulated by replacing homotopy with homology: a group is said to be of type FHn if it acts as above on a CW-complex whose n first homology groups vanish.

Algebraic finiteness properties
Let $$\Gamma$$ be a group and $$\mathbb Z\Gamma$$ its group ring. The group $$\Gamma$$ is said to be of type FPn if there exists a resolution of the trivial $$\mathbb Z\Gamma$$-module $$\mathbb Z$$ such that the n first terms are finitely generated projective $$\mathbb Z\Gamma$$-modules. The types FP∞ and FP are defined in the obvious way.

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

It is also possible to define classes FPn(R) and FLn(R) for any commutative ring R, by replacing the group ring $$\mathbb Z\Gamma$$ by $$R\Gamma$$ in the definitions above.

Either of the conditions Fn or FHn imply FPn and FLn (over any commutative ring). A group is of type FP1 if and only if it is finitely generated, but for any n ≥ 2 there exists groups which are of type FPn but not Fn.

Group cohomology
If a group is of type FPn then its cohomology groups $$H^i(\Gamma)$$ are finitely generated for $$0 \le i \le n$$. 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.

Finite groups
A finite cyclic group $$G$$ acts freely on the unit sphere in $$\mathbb R^{\mathbb N}$$, preserving a CW-complex structure with finitely many cells in each dimension. Since this unit sphere is contractible, every finite cyclic group is of type F∞.

The standard resolution for a group $$G$$ gives rise to a contractible CW-complex with a free $$G$$-action in which the cells of dimension $$n$$ correspond to $$(n+1)$$-tuples of elements of $$G$$. This shows that 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 $$\Gamma$$ is a torsion-free, finitely generated nilpotent group then it is of type F.

Geometric conditions for finiteness properties
Negatively curved groups (hyperbolic or CAT(0) groups) are always of type F∞. 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 $$\Gamma$$ is an arithmetic group in a simple algebraic group of rank $$r$$ over a global function field (such as $$\mathbb F_q(t)$$) then it is of type Fr but not of type Fr+1.