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- W4311554500 abstract "Let N be a finitely generated nilpotent group. The subgroup zeta function $$zeta _N^{scriptscriptstyle leqslant }(s)$$ and the normal zeta function $$zeta _N^{scriptscriptstyle lhd }(s)$$ of N are Dirichlet series enumerating the finite index subgroups or the finite index normal subgroups of N. We present results about their abscissae of convergence $$alpha _N^{scriptscriptstyle leqslant }$$ and $$alpha _N^{scriptscriptstyle lhd }$$ , also known as the degrees of polynomial subgroup growth and polynomial normal subgroup growth of N, respectively. We first prove some upper bounds for the functions $$Nmapsto alpha _N^{scriptscriptstyle leqslant }$$ and $$Nmapsto alpha _N^{scriptscriptstyle lhd }$$ when restricted to the class of torsion-free nilpotent groups of a fixed Hirsch length. We then show that if two finitely generated nilpotent groups have isomorphic $$mathbb {C}$$ -Mal’cev completions, then their subgroup (resp. normal) zeta functions have the same abscissa of convergence. This follows, via the Mal’cev correspondence, from a similar result that we establish for zeta functions of rings. This result is obtained by proving that the abscissa of convergence of an Euler product of certain Igusa-type local zeta functions introduced by du Sautoy and Grunewald remains invariant under base change. We also apply this methodology to formulate and prove a version of our result about nilpotent groups for virtually nilpotent groups. As a side application of our result about zeta functions of rings, we present a result concerning the distribution of orders in number fields." @default.
- W4311554500 created "2022-12-27" @default.
- W4311554500 creator A5069979923 @default.
- W4311554500 date "2022-12-05" @default.
- W4311554500 modified "2023-10-14" @default.
- W4311554500 title "On the degree of polynomial subgroup growth of nilpotent groups" @default.
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- W4311554500 doi "https://doi.org/10.1007/s00209-022-03156-8" @default.
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