Subgroup growth

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In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group.[1]

Let G be a finitely generated group. Then, for each integer n define n(G) to be the number of subgroups U of index n in G. Similarly, if G is a topological group, s_n(G) denotes the number of open subgroups U of index n in G. One similarly defines m_n(G) and to denote the number of maximal and normal subgroups of index n, 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[edit]

Let G 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).

Zn → G

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 ps.

As an example, let G be the discrete Heisenberg group. This group has a "presentation" with generators x, y, z and relations

Hence, elements of G can be represented as triples (a, b, c) of integers with group operation given by

To each finite index subgroup U of G, associate the set of all ``good bases´´ of U as follows. Note that G has a normal series

with infinite cyclic factors. A triple (g_1, g_2, g_3) in G is called a good basis of U, if g_1, g_2, g_3 generate U, and . In general, it is quite complicated to determine the set of good bases for a fixed subgroup U. 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 μ is the Haar measure on Zp, denotes the p-adic absolute value and is the set of tuples of p-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 ζG(s) 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 ζG(s). The local factor

can always be expressed as a definable p-adic integral. Applying a result of MacIntyre on the model theory of p-adic integers, one deduces again that ζG(s) is a rational function in ps. 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 ζG(s) is meromorphic in the domain

Re s > α − δ,

where α is the abscissa of convergence of ζG(s), and δ is some positive number, and holomorphic in some neighbourhood of . Using a Tauberian theorem this implies

for some real number α and a non-negative integer k.

Congruence subgroups[edit]

Subgroup growth and coset representations[edit]

Let G be a group, U a subgroup of index n. Then G acts on the set of left cosets of U in G by left shift:

In this way, U induces a homomorphism of G into the symmetric group on . G acts transitively on , and vice versa, given a transitive action of G on

the stabilizer of the point 1 is a subgroup of index n in G. Since the set

can be permuted in

ways, we find that is equal to the number of transitive G-actions divided by . Among all G-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.

References[edit]

  1. ^ Alexander Lubotzky, Dan Segal (2003). Subgroup Growth. Birkhäuser. ISBN 3-7643-6989-2.