Subgroup growth

In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group.
Let formula_1 be a finitely generated group. Then, for each integer formula_2 define formula_3 to be the number of subgroups formula_4 of index formula_2 in formula_1. Similarly, if formula_1 is a topological group, formula_8 denotes the number of open subgroups formula_9 of index formula_2 in formula_1. One similarly defines formula_12 and formula_13 to denote the number of maximal and normal subgroups of index formula_2, 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 formula_15 be a finitely generated torsionfree nilpotent group. Then there exists a composition series with infinite cyclic factors, which induces a bijection (not though necessarily a homomorphism).
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 formula_18.
As an example, let formula_19 be the discrete Heisenberg group. This group has a "presentation" with generators formula_20 and relations
Hence, elements of formula_19 can be represented as triples formula_23 of integers with group operation given by
To each finite index subgroup formula_25 of formula_19, associate the set of all "good bases" of formula_27 as follows. Note that formula_19 has a normal series
with infinite cyclic factors. A triple formula_30 is called a "good basis" of formula_25, if formula_32 generate formula_25, and formula_34. In general, it is quite complicated to determine the set of good bases for a fixed subgroup formula_25. 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 formula_37 is the Haar measure on formula_38, formula_39 denotes the p-adic absolute value and formula_40 is the set of tuples of formula_41-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 formula_46 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 formula_48. The local factor
can always be expressed as a definable formula_41-adic integral. Applying a result of MacIntyre on the model theory of formula_51-adic integers, one deduces again that formula_52 is a rational function in formula_18. 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 formula_54, they showed that for any torsionfree nilpotent group, the function formula_48 is meromorphic in the domain
where formula_57 is the abscissa of convergence of formula_52, and formula_59 is some positive number, and holomorphic in some neighbourhood of formula_60. Using a Tauberian theorem this implies
for some real number formula_57 and a non-negative integer formula_63.
Subgroup growth and coset representations.
Let formula_19 be a group, formula_25 a subgroup of index formula_66. Then formula_19 acts on the set of left cosets of formula_27 in formula_69 by left shift:
In this way, formula_25 induces a homomorphism of formula_19 into the symmetric group on formula_73. formula_69 acts transitively on formula_73, and vice versa, given a transitive action of formula_19 on
the stabilizer of the point 1 is a subgroup of index formula_66 in formula_19. Since the set
can be permuted in
ways, we find that formula_8 is equal to the number of transitive formula_69-actions divided by formula_81. Among all formula_15-actions, we can distinguish transitive actions by a sifting argument, to arrive at the following formula
where formula_87 denotes the number of homomorphisms
In several instances the function formula_87 is easier to be approached then formula_8, and, if formula_87 grows sufficiently large, the sum is of negligible order of magnitude, hence, one obtains an asymptotic formula for formula_8.
As an example, let formula_93 be the free group on two generators. Then every map of the generators of formula_93 extends to a homomorphism
that is
From this we deduce
For more complicated examples, the estimation of formula_87 involves the representation theory and statistical properties of symmetric groups.