Subgroup growth


In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group.
Let be a finitely generated group. Then, for each integer define to be the number of subgroups of index in. Similarly, if is a topological group, denotes the number of open subgroups of index in. One similarly defines and to denote the number of maximal and normal subgroups of index, respectively.
Subgroup growth studies these functions, their interplay, and the characterization of group theoretical properties in terms of these functions.
The theory was motivated by the desire to enumerate finite groups of given order, and the analogy with Mikhail Gromov's notion of word growth.

Nilpotent groups

Let be a finitely generated torsionfree nilpotent group. Then there exists a composition series with infinite cyclic factors, which induces a bijection.
such that group multiplication can be expressed by polynomial functions in these coordinates; in particular, the multiplication is definable. Using methods from the model theory of p-adic integers, F. Grunewald, D. Segal and G. Smith showed that the local zeta function
is a rational function in.
As an example, let be the discrete Heisenberg group. This group has a "presentation" with generators and relations
Hence, elements of can be represented as triples of integers with group operation given by
To each finite index subgroup of, associate the set of all "good bases" of as follows. Note that has a normal series
with infinite cyclic factors. A triple is called a good basis of, if generate, and. In general, it is quite complicated to determine the set of good bases for a fixed subgroup. To overcome this difficulty, one determines the set of all good bases of all finite index subgroups, and determines how many of these belong to one given subgroup. To make this precise, one has to embed the Heisenberg group over the integers into the group over p-adic numbers. After some computations, one arrives at the formula
where is the Haar measure on, denotes the p-adic absolute value and is the set of tuples of -adic integers
such that
is a good basis of some finite-index subgroup. The latter condition can be translated into
Now, the integral can be transformed into an iterated sum to yield
where the final evaluation consists of repeated application of the formula for the value of the geometric series. From this we deduce that can be expressed in terms of the Riemann zeta function as
For more complicated examples, the computations become difficult, and in general one cannot expect a closed expression for. The local factor
can always be expressed as a definable -adic integral. Applying a result of MacIntyre on the model theory of -adic integers, one deduces again that is a rational function in. Moreover, M. du Sautoy and F. Grunewald showed that the integral can be approximated by Artin L-functions. Using the fact that Artin L-functions are holomorphic in a neighbourhood of the line, they showed that for any torsionfree nilpotent group, the function is meromorphic in the domain
where is the abscissa of convergence of, and is some positive number, and holomorphic in some neighbourhood of. Using a Tauberian theorem this implies
for some real number and a non-negative integer.

Congruence subgroups

Subgroup growth and coset representations

Let be a group, a subgroup of index. Then acts on the set of left cosets of in by left shift:
In this way, induces a homomorphism of into the symmetric group on. acts transitively on, and vice versa, given a transitive action of on
the stabilizer of the point 1 is a subgroup of index in. Since the set
can be permuted in
ways, we find that is equal to the number of transitive -actions divided by. Among all -actions, we can distinguish transitive actions by a sifting argument, to arrive at the following formula
where denotes the number of homomorphisms
In several instances the function is easier to be approached then, and, if grows sufficiently large, the sum is of negligible order of magnitude, hence, one obtains an asymptotic formula for.
As an example, let be the free group on two generators. Then every map of the generators of extends to a homomorphism
that is
From this we deduce
For more complicated examples, the estimation of involves the representation theory and statistical properties of symmetric groups.