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# 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 $a_{n}^{\leqslant}(G)=a_{n}(\mathcal{S}(G))$ and $a_{n}^{\trianglelefteqslant}(G)=a_{n}(\mathcal{N}(G))$ and define

$\zeta_{G}^{\leqslant}(s)=\zeta_{{\mathcal{S}(G)}}(s)=\sum_{{H\leqslant_{{% \mathrm{f}}}G}}|G:H|^{{-s}},$ |

and

$\zeta_{G}^{\trianglelefteqslant}(s)=\zeta_{{\mathcal{N}(G)}}(s)=\sum_{{N% \trianglelefteqslant_{{\mathrm{f}}}G}}|G:N|^{{-s}}.$ |

If, in addition, $G$ is nilpotent, then $\zeta_{G}^{\leqslant}$ has a decomposition as a formal Euler product

$\zeta_{G}^{\leqslant}(s)=\prod_{{p\text{ prime}}}\zeta_{{G,p}}^{\leqslant}(s),$ |

where

$\zeta_{{G,p}}^{\leqslant}(s)=\sum_{{i=0}}^{\infty}a_{{p^{i}}}^{\leqslant}(G)p^% {{-is}}.$ |

An analogous result holds for the normal zeta function $\zeta_{G}^{\trianglelefteqslant}$. The result for both $\zeta_{G}^{\leqslant}$ and $\zeta_{G}^{\trianglelefteqslant}$ 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 $\zeta_{{G,p}}^{\leqslant}(s)$ and $\zeta_{{G,p}}^{\trianglelefteqslant}(s)$ 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 $\alpha_{G}^{\leqslant}$ to be the abscissa of convergence of $\zeta_{G}^{\leqslant}$. That is, $\alpha_{G}^{\leqslant}$ is the smallest $\alpha\in\mathbb{R}$ such that $\zeta_{G}^{\leqslant}$ defines a holomorphic function in the right half-plane $\{z\in\mathbb{C}\mid\Re(z)>\alpha\}$. It can then be shown that $\alpha_{G}^{\leqslant}\leqslant\mathrm{h}(G)$, where $\mathrm{h}(G)$ is the Hirsch number of $G$. Therefore, if $G$ is a $\mathcal{T}$-group, $\zeta_{G}^{\leqslant}$ 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.

## Mathematics Subject Classification

20E07*no label found*20F69

*no label found*20F18

*no label found*

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