Zeta function of a group

Let $G$ be a finitely generated group and let $\mathcal{X}$ be a family of finite index subgroups of $G$ . Define$$ a_n(\mathcal{X}) = |\{H \in \mathcal{X} \mid |G : H| = n\}|.$$ 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 $\mathcal{X}$ to be the formal Dirichlet series$$ \zeta_{\mathcal{X}}(s) = \sum_{n=1}^\infty a_n(\mathcal{X}) n^{-s}.$$

Two important special cases are the zeta function counting all subgroups and the zeta function counting normal subgroups. Let $\mathcal{S}(G)$ and $\mathcal{N}(G)$ be the families of all finite index subgroups of $G$ and of all finite index normal subgroups of $G$ , respectively. We write $a_n^\sgp(G) = a_n(\mathcal{S}(G))$ and $a_n^\nsgp(G) = a_n(\mathcal{N}(G))$ and define$$ \zeta_G^\sgp(s) = \zeta_{\mathcal{S}(G)}(s) = \sum_{H \sgpf G} |G : H|^{-s},$$ and$$ \zeta_G^\nsgp(s) = \zeta_{\mathcal{N}(G)}(s) = \sum_{N \nsgpf G} |G : N|^{-s}.$$

If, in addition, $G$ is nilpotent, then $\zeta_G^\sgp$ has a decomposition as a formal Euler product$$ \zeta_G^\sgp(s) = \prod_{p \text{ prime}} \zeta_{G,p}^\sgp(s),$$ where$$ \zeta_{G,p}^\sgp(s) = \sum_{i=0}^\infty a_{p^i}^\sgp(G) p^{-is}.$$ An analogous result holds for the normal zeta function $\zeta_G^\nsgp$ . The result for both $\zeta_G^\sgp$ and $\zeta_G^\nsgp$ can be proved using properties of the profinite completion of $G$ . 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 $p$ -adic integrals, which can in turn be used to prove (using some high-powered machinery) that $\zeta_{G,p}^\sgp(s)$ and $\zeta_{G,p}^\nsgp(s)$ are rational functions in $p$ and $p^{-s}$ .

In the case when $G$ is a $\mathcal{T}$ -group, that is, $G$ is finitely generated, torsion free, and nilpotent, define $\alpha_G^\sgp$ to be the abscissa of convergence of $\zeta_G^\sgp$ . That is, $\alpha_G^\sgp$ is the smallest $\alpha \in \mathbb{R}$ such that $\zeta_G^\sgp$ defines a holomorphic function in the right half-plane $\{z \in \mathbb{C} \mid \Re(z) > \alpha\}$ . It can then be shown that $\alpha_G^\sgp \leqslant \hir(G)$ , where $\hir(G)$ is the Hirsch number of $G$ . Therefore, if $G$ is a $\mathcal{T}$ -group, $\zeta_G^\sgp$ defines a holomorphic function in some right half-plane.

Bibliography

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.