Finiteness Properties Of Groups
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, finiteness properties of a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
are a collection of properties that allow the use of various algebraic and
topological Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
tools, for example
group cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, 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 An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
''n'' ≥ 1, a group \Gamma is said to be ''of type'' ''F''''n'' if there exists an aspherical CW-complex whose
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to \Gamma (a
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e., a topological space all of whose homotopy groups are trivial) by a proper free ...
for \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 ''F''1 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 In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathematics), group. Its definition is sug ...
of the group for this generating family is the 1-skeleton of its universal cover); * a group is of type ''F''2 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 F 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 The Acts of the Apostles (, ''Práxeis Apostólōn''; ) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message to the Roman Empire. Acts and the Gospel of Luke make up a two-par ...
properly discontinuously, freely and cocompactly on a CW-complex whose
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
s \pi_0, \ldots, \pi_ 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 \Gamma be a group and \mathbb Z\Gamma its
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...
. The group \Gamma is said to be of type FP''n'' 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 module In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...
s 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 In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
''R'', by replacing the group ring \mathbb Z\Gamma by 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 ''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''. If a group is of type ''F''''2'' and ''FP''''n'', then it is of type ''F''''n''.


Group cohomology

If a group is of type ''FP''''n'' 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.


Examples


Finite groups

A finite cyclic group G acts freely on the unit sphere in \mathbb R^, 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 In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or a particularly simple object possessing a given structure (e.g., group, topological space). The noun triviality usual ...
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 In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group ...
is never of type ''F''.


Nilpotent groups

If \Gamma is a torsion-free, finitely generated
nilpotent group In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, it has a central series of finite length or its lower central series terminates with . I ...
then it is of type F.


Geometric conditions for finiteness properties

Negatively curved groups (
hyperbolic Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...
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 group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
s over
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s are of type F. The Borel–Serre compactification shows that this is also the case for non-cocompact arithmetic groups. Arithmetic groups over
function field Function field may refer to: *Function field of an algebraic variety *Function field (scheme theory) *Algebraic function field *Function field sieve *Function field analogy Function or functionality may refer to: Computing * Function key, a ty ...
s have very different finiteness properties: if \Gamma is an arithmetic group in a simple algebraic group of
rank A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * Diplomatic rank * Hierarchy ...
r over a global function field (such as \mathbb F_q(t)) then it is of type Fr but not of type Fr+1.


Notes


References

* * {{refend Group theory Homological algebra Geometric group theory