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 integern ≥ 1, a group is said to be of typeFn if there exists an aspherical CW-complex whose fundamental group is isomorphic to 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 F1if and only if it is finitely generated ;
a group is of type F2 if and only if it is finitely presented.
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 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 actsproperly 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 FHn 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 FPn if there exists a resolution of the trivial -module such that the n first terms are finitely generated projective -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 and FLn for any commutative ringR, by replacing the group ring by in the definitions above. Either of the conditions Fn or FHn imply FPn and FLn. 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 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. 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.
Negatively curved 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 is an arithmetic group in a simple algebraic group of rank over a global function field then it is of type Fr but not of type Fr+1.
