Isomorphism classes and zeta-functions of some nilpotent groups. II.

*(English)*Zbl 1345.20030Let \(G\) be a finitely generated group. The subgroup zeta function of \(G\), denoted by \(\zeta_G(s)\), is defined as the Dirichlet series associated to the sequence \(\{a_n(G)\}_n\), where \(a_n(G)\) is the number of subgroups of \(G\) of index \(n\). Zeta functions of groups were introduced by F. J. Grunewald, D. Segal and G. C. Smith: they proved in particular that \(\zeta_G(s)\) has remarkable properties when \(G\) belongs to the class of torsion-free finitely generated nilpotent groups (\(\mathcal T\)-groups). It is known that for the class of all \(\mathcal T\)-groups, the isomorphim classes are not determined by zeta functions. However this article gives a class of \(\mathcal T\)-groups with the property that if \(G_1\) and \(G_2\) belong to this class, then \(\zeta_{G_1}(s)=\zeta_{G_2}(s)\) if and only if \(G_1\) and \(G_2\) are isomorphic.

For part I see [the author, Tokyo J. Math. 36, No. 1, 163-175 (2013; Zbl 1287.20035)].

For part I see [the author, Tokyo J. Math. 36, No. 1, 163-175 (2013; Zbl 1287.20035)].

Reviewer: Andrea Lucchini (Padova)

##### MSC:

20E07 | Subgroup theorems; subgroup growth |

20F18 | Nilpotent groups |

11M41 | Other Dirichlet series and zeta functions |

##### Keywords:

zeta functions; finitely generated nilpotent groups; subgroup growth; subgroups of finite index**OpenURL**

##### References:

[1] | F. J. Grunewald and R. Scharlau, A note on finitely generated torsion-free groups of class 2, J. Algebra 58 (1979), 162-175. · Zbl 0406.20031 |

[2] | F. J. Grunewald, D. Segal and G. C. Smith, Subgroups of finite index in nilpotent groups, Invent. Math. 93 (1988), 185-223. · Zbl 0651.20040 |

[3] | F. Hyodo, Isomorphism classes and Zeta-functions of some Nilpotent Groups, Tokyo J. Math. 36 (2013), 163-175. · Zbl 1287.20035 |

[4] | P. F. Pickel, Finitely generated nilpotent groups with isomorphic finite quotients, Trans. Amer. Math. Soc. 160 (1971), 327-341. · Zbl 0235.20027 |

[5] | C. Voll, Functional equations for zeta functions of groups and rings, Ann. of Math. 172 (2010), no. 2, 1181-1218. · Zbl 1314.11057 |

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