Analytic properties of zeta functions and subgroup growth

2000/11/01 by Marcus du Sautoy, Fritz Grunewald · 1 citation
Mathematics · #math.GR

paper · pdf

published as Ann. of Math. (2) 152 (2000), no. 3, 793--833 · 41 pages, published version, abstract added in migration

arxiv created 2000/11/01 · arxiv updated 2009/11/30

Abstract

In this paper we introduce some new methods to understand the analytic behaviour of the zeta function of a group. We can then combine this knowledge with suitable Tauberian theorems to deduce results about the growth of subgroups in a nilpotent group. In order to state our results we introduce the following notation. For αa real number and N a nonnegative integer, define s_N^α(G) = sum_n=1^N a_n(G)/n^α. Main Theorem: Let G be a finitely generated nilpotent infinite group. (1) The abscissa of convergence α(G) of ζ_G(s) is a rational number and ζ_G(s) can be meromorphically continued to Re(s)>α(G)-δfor some δ>0. The continued function is holomorphic on the line \Re(s) = (α)G except for a pole at s=α(G). (2) There exist a nonnegative integer b(G) and some real numbers c,c' such that s_N(G) ~ c N^α(G)(log N)^b(G) s_N^α(G)(G) ~ c' (log N)^b(G)+1 for N→ ∞ .

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