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 ${\displaystyle \Gamma }$ is said to be of typeF_n if there exists an aspherical CW-complex whose fundamental group is isomorphic to ${\displaystyle \Gamma }$ (a classifying space for ${\displaystyle \Gamma }$) 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 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 F_n which are not of type F_n+1. Finite groups are of type F_∞ but not of type F. Thompson's group ${\displaystyle F}$ is an example of a torsion-free group which is of type F_∞ but not of type F. ^[1]

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 ${\displaystyle \pi _{1},\ldots ,\pi _{n}}$ 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 ${\displaystyle \Gamma }$ be a group and ${\displaystyle \mathbb {Z} \Gamma }$ its group ring. The group ${\displaystyle \Gamma }$ is said to be of type FP_n if there exists a resolution of the trivial ${\displaystyle \mathbb {Z} \Gamma }$-module ${\displaystyle \mathbb {Z} }$ such that the n first terms are finitely generated projective ${\displaystyle \mathbb {Z} \Gamma }$-modules. ^[2] 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 ${\displaystyle \mathbb {Z} \Gamma }$ by ${\displaystyle R\Gamma }$ 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 FP₁ if and only if it is finitely generated, ^[2] but for any n ≥ 2 there exists groups which are of type FP_n but not F_n. ^[3]

Group cohomology

Main article: Group cohomology

If a group is of type FP_n then its cohomology groups ${\displaystyle H^{i}(\Gamma )}$ are finitely generated for ${\displaystyle 0\leq i\leq n}$. 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.

Examples

Finite groups

A finite cyclic group ${\displaystyle G}$ acts freely on the unit sphere in ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$, preserving a CW-complex structure with finitely many cells in each dimension. ^[4] Since this unit sphere is contractible, every finite cyclic group is of type F_∞.

The standard resolution ^[5] for a group ${\displaystyle G}$ gives rise to a contractible CW-complex with a free ${\displaystyle G}$-action in which the cells of dimension ${\displaystyle n}$ correspond to ${\displaystyle (n+1)}$-tuples of elements of ${\displaystyle G}$. 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 ${\displaystyle \Gamma }$ is a torsion-free, finitely generated nilpotent group then it is of type F. ^[6]

Geometric conditions for finiteness properties

Negatively curved groups (hyperbolic or CAT(0) groups) are always of type F_∞. ^[7] 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 ${\displaystyle \Gamma }$ is an arithmetic group in a simple algebraic group of rank ${\displaystyle r}$ over a global function field (such as ${\displaystyle \mathbb {F} _{q}(t)}$) then it is of type F_r but not of type F_r+1. ^[8]

Notes

1. Brown, Kenneth; Geoghegan, Ross (1984). "An infinite-dimensional torsion-free FP_∞ group". Inventiones Mathematicae. 77 (2): 367–381. doi:10.1007/BF01388451. MR   0752825. S2CID   121877111.
2. Brown 1982, p. 197.
3. 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, S2CID   120422255
4. Brown 1982, p. 20.
5. Brown 1982, p. 18.
6. Brown 1982, p. 213.
7. Bridson & Haefliger 1999, p. 439, 468.
8. 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. S2CID   53991649.

References

• Bridson, Martin; Haefliger, André (1999). Metric spaces of non-positive curvature. Springer-Verlag. ISBN   3-540-64324-9.
• Brown, Kenneth S. (1982). Cohomology of groups. Springer-Verlag. ISBN   0-387-90688-6.