Zeta function of a group
Let be a finitely generated group and let be a
family of finite index subgroups of . Define
Note that these numbers are finite since a finitely generated group
has only finitely many subgroups of a given index. We define the
zeta function of the family to be the formal
Dirichlet series
Two important special cases are the zeta function counting all
subgroups and the zeta function counting normal subgroups. Let
and be the families of all finite
index subgroups of and of all finite index normal subgroups of
, respectively. We write and
and define
and
If, in addition, is nilpotent, then has a
decomposition as a formal Euler product
where
An analogous result holds for the normal zeta function
. The result for both and
can be proved using properties of the profinite
completion of . However, a simpler proof for the normal zeta
function is provided by the fact that a finite nilpotent group
decomposes into a direct product of its Sylow subgroups. These results
allow the zeta functions to be expressed in terms of -adic
integrals, which can in turn be used to prove (using some high-powered
machinery) that and are
rational functions in and .
In the case when is a -group, that is, is
finitely generated, torsion free, and nilpotent, define
to be the abscissa of convergence of . That is,
is the smallest such that
defines a holomorphic function
in the right half-plane
. It can then be shown that
, where is the Hirsch number
of . Therefore, if is a -group,
defines a holomorphic function in some right half-plane.
References
- 1 F. J. Grunewald, D. Segal, and G. C. Smith, Subgroups of finite index in nilpotent groups, Invent. math. 93 (1988), 185–223.
- 2 M. P. F. du Sautoy, Zeta functions of groups: the quest for order versus the flight from ennui, Groups St. Andrews 2001 in Oxford. Vol. I, London Math. Soc. Lecture Note Ser., vol. 304, Cambridge Univ. Press, 2003, pp. 150–189.
Title | Zeta function of a group |
---|---|
Canonical name | ZetaFunctionOfAGroup |
Date of creation | 2013-03-22 15:16:00 |
Last modified on | 2013-03-22 15:16:00 |
Owner | avf (9497) |
Last modified by | avf (9497) |
Numerical id | 7 |
Author | avf (9497) |
Entry type | Definition |
Classification | msc 20E07 |
Classification | msc 20F69 |
Classification | msc 20F18 |
Related topic | Group |