Editing Subgroup growth (section)

==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>.