Editing Subgroup growth

In [[mathematics]], '''subgroup growth''' is a branch of [[group theory]], dealing with quantitative questions about [[subgroup]]s of a given [[group (mathematics)|group]].<ref>{{cite book|title=Subgroup Growth|author=[[Alexander Lubotzky]], Dan Segal|year=2003|publisher=Birkhäuser|isbn=3-7643-6989-2}}</ref>

Let <math>G</math> be a [[finitely generated group]]. Then, for each integer <math>n</math> define <math>a_n(G)</math> to be the number of subgroups <math>H</math> of [[Index of a subgroup|index]] <math>n</math> in <math>G</math>. Similarly, if <math>G</math> is a [[topological group]], <math>s_n(G)</math> denotes the number of open subgroups <math>U</math> of index <math>n</math> in <math>G</math>. One similarly defines  <math>m_n(G)</math> and <math>s_n^\triangleleft(G)</math> to denote the number of [[maximal subgroup|maximal]] and [[normal subgroup]]s of index <math>n</math>, 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 (mathematician)|Mikhail Gromov]]'s notion of [[Gromov's theorem on groups of polynomial growth|word growth]].

==Nilpotent groups==
Let <math>G </math> be a finitely generated [[Torsion subgroup|torsionfree]] [[nilpotent group]]. Then there exists a [[composition series]] with infinite [[Cyclic group|cyclic]] factors, which induces a bijection (though not necessarily a [[Group homomorphism|homomorphism]]).

:<math>\mathbb{Z}^n \longrightarrow G </math>

such that group multiplication can be expressed by polynomial functions in these coordinates; in particular, the multiplication is [[First-order predicate calculus|definable]]. Using methods from the [[model theory]] of [[p-adic integer]]s, F. Grunewald, D. Segal and G. Smith showed that the [[local zeta function]]
:<math>
\zeta_{G, p}(s) = \sum_{\nu=0}^\infty s_{p^n}(G) p^{-ns}
</math>
is a [[rational function]] in <math>p^{-s} </math>.

As an example, let <math> G </math> be the discrete [[Heisenberg group]]. This group has a "presentation" with [[generating set of a group|generators]] <math>x, \, y, \, z </math> and [[presentation of a group|relations]]
:<math>
[x, y] = z, [x, z] = [y, z] = 1.
</math>
Hence, elements of <math> G </math> can be represented as triples <math> (a,\, b, \, c) </math> of integers with group operation given by
:<math>
(a, b, c)\circ(a', b', c') = (a+a', b+b', c+c'+ab').
 </math>
To each finite index [[subgroup]] <math> U </math> of <math> G </math>, associate the [[Set (mathematics)|set]] of all "good bases" of <math> U</math> as follows. Note that <math> G </math> has a [[normal series]]
:<math>
G=\langle x, y, z\rangle\triangleright\langle y, z\rangle\triangleright\langle z\rangle\triangleright 1
</math>
with infinite [[cyclic number (group theory)|cyclic]] [[divisor|factors]]. A triple <math>(g_1, g_2, g_3) \in G </math> is called a ''good basis'' of <math> U </math>, if <math>g_1, g_2, g_3 </math> generate <math> U </math>, and <math>g_2\in\langle y, z\rangle, g_3\in\langle z\rangle</math>. In general, it is quite complicated to determine the set of good bases for a fixed subgroup <math> U </math>. 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 number]]s. After some computations, one arrives at the formula
:<math>
\zeta_{G, p}(s) = \frac{1}{(1-p^{-1})^3}\int_\mathcal{M} |a_{11}|_p^{s-1} |a_{22}|_p^{s-2} |a_{33}|_p^{s-3}\;d\mu,
</math>
where <math>\mu </math> is the [[Haar measure]] on <math>\mathbb{Z}_p </math>, <math>|\cdot|_p</math> denotes the [[P-adic number|p-adic absolute value]] and <math>\mathcal{M}</math> is the set of tuples of <math> p </math>-adic integers
:<math>
\{a_{11}, a_{12}, a_{13}, a_{22}, a_{23}, a_{33}\}
</math>
such that
:<math>
\{x^{a_{11}}y^{a_{12}}z^{a_{13}}, y^{a_{22}}z^{a_{23}}, z^{a_{33}}\}
</math>
is a good basis of some finite-index subgroup. The latter condition can be translated into

:<math>a_{33}|a_{11}\cdot a_{22}</math>.

Now, the integral can be transformed into an iterated sum to yield
:<math>
\zeta_{G, p}(s) = \sum_{a\geq 0}\sum_{b\geq 0}\sum_{c=0}^{a+b} p^{-as-b(s-1)-c(s-2)} = \frac{1-p^{3-3s}}{(1-p^{-s})(1-p^{1-s})(1-p^{2-2s})(1-p^{2-3s})}
</math>
where the final evaluation consists of repeated application of the formula for the value of the [[geometric series]]. From this we deduce that <math>\zeta_G (s) </math> can be expressed in terms of the [[Riemann zeta function]] as
:<math>
\zeta_G(s) = \frac{\zeta(s)\zeta(s-1)\zeta(2s-2)\zeta(2s-3)}{\zeta(3s-3)}.
</math>

For more complicated examples, the computations become difficult, and in general one cannot expect a [[closed expression]] for <math> \zeta_G(s)</math>. The local factor

:<math>\zeta_{G, p}(s)</math>

can always be expressed as a definable <math> p </math>-adic integral. Applying a result of [[MacIntyre]] on the model theory of <math> p</math>-adic integers, one deduces again that <math>\zeta_G(s) </math> is a rational function in <math>p^{-s} </math>. Moreover, [[M. du Sautoy]] and F. Grunewald showed that the integral can be approximated by [[Artin L-function]]s. Using the fact that Artin L-functions are holomorphic in a neighbourhood of the line <math>\Re (s)=1</math>, they showed that for any torsionfree nilpotent group, the function <math> \zeta_G(s)</math> is [[meromorphic]] in the domain

:<math>\Re(s)>\alpha-\delta </math>

where <math>\alpha </math> is the [[abscissa of convergence]] of <math>\zeta_G(s) </math>, and <math> \delta </math> is some positive number, and holomorphic in some neighbourhood of <math>\Re (s)=\alpha</math>. Using a [[Tauberian theorem]] this implies
:<math>
\sum_{n\leq x} s_n(G) \sim x^\alpha\log^k x
</math>
for some real number <math>\alpha </math> and a non-negative integer <math> k </math>.

==Congruence subgroups==
{{Empty section|date=July 2010}}

==Subgroup growth and coset representations==
Let <math> G </math> be a group, <math> U </math> a subgroup of index <math> n</math>. Then <math> G </math> acts on the set of left [[coset]]s of <math> U</math> in <math> G</math> by left shift:

:<math>g(hU)=(gh)U.</math>

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

:<math>\{1, \ldots, n\},</math>

the stabilizer of the point 1 is a subgroup of index <math> n</math> in <math> G </math>. Since the set

:<math>\{2, \ldots, n\}</math>

can be permuted in

:<math>(n-1)!</math>

ways, we find that <math>s_n(G)</math> is equal to the number of transitive [[Group action (mathematics)|<math> G</math>-actions]] divided by <math>(n-1)!</math>. Among all <math>G </math>-actions, we can distinguish transitive actions by a [[sifting argument]], to arrive at the following formula

:<math>
s_n(G) = \frac{h_n(G)}{(n-1)!} - \sum_{\nu=1}^{n-1} \frac{h_{n-\nu}(G)s_\nu(G)}{(n-\nu)!},
</math>

where <math>h_n(G)</math> denotes the number of homomorphisms

:<math>\varphi:G\rightarrow S_n.</math>

In several instances the function <math>h_n(G)</math> is easier to be approached then <math>s_n(G)</math>, and, if <math>h_n(G)</math> grows sufficiently large, the sum is of negligible order of magnitude, hence, one obtains an [[asymptotic expansion|asymptotic]] formula for <math>s_n(G)</math>.

As an example, let <math>F_2</math> be the [[free group]] on two generators. Then every map of the generators of <math>F_2</math> extends to a homomorphism

:<math>F_2\rightarrow S_n,</math>

that is

:<math>h_n(F_2)=(n!)^2.</math>

From this we deduce

:<math>s_n(F_2)\sim n\cdot n!.</math>

For more complicated examples, the estimation of <math>h_n(G)</math> involves the [[representation theory]] and [[statistical properties of symmetric groups]].

==References==
{{reflist}}

[[Category:Infinite group theory]]
[[Category:Zeta and L-functions]]