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 formula_1 is said to be "of type" "F""n" if there exists an aspherical CW-complex whose fundamental group is isomorphic to formula_1 (a classifying space for formula_1) 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: 
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". Thompson's group formula_4 is an example of a torsion-free group which is 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 formula_5 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 formula_1 be a group and formula_7 its group ring. The group formula_1 is said to be of type FP"n" if there exists a resolution of the trivial formula_7-module formula_10 such that the "n" first terms are finitely generated projective formula_7-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 "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 formula_7 by formula_13 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, but for any "n" ≥ 2 there exists groups which are of type "FP""n" but not "F""n".
Group cohomology.
If a group is of type "FP""n" then its cohomology groups formula_14 are finitely generated for formula_15. 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 formula_16 acts freely on the unit sphere in formula_17, 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". 
Nilpotent groups.
If formula_1 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 formula_1 is an arithmetic group in a simple algebraic group of rank formula_20 over a global function field (such as formula_21) then it is of type Fr but not of type Fr+1. 