Analytic properties of zeta functions and subgroup growth

By Marcus du Sautoy, Fritz Grunewald. 2000

127 citations recorded by the source corpus.

It has become somewhat of a cottage industry over the last fifteen years to understand the rate of growth of the number of subgroups of finite index in a group G. Although the story began much before, the recent activity grew out of a paper by Dan Segal in [36]. The story so far has been well-documented in Lubotzky’s subsequent survey paper in [30]. In [24] the second author of this article, Segal and Smith introduced the zeta function of a group as a tool for understanding this growth of subgroups. Let an(G) be the number of subgroups of index n in the finitely generated group G and sN (G) = a1(G) + · · ·+ aN (G) be the number of subgroups of index N or less. The zeta function is defined as the Dirichlet series with coefficients an(G) and has a natural interpretation as a noncommutative generalization of the Dedekind zeta function of a number field:

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a73b7d31bf4a70ca2b82f5779985dddf766f3cbc

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B (2011, background): “Is the abscissa of convergence α(G(OL)) independent of L? Is it a rational number? Does the zeta function ζG(O)(s) allow for analytic continuation beyond its abscissa of convergence? The last two questions would have an affirmative answer if local representation zeta functions of T -groups were ‘con”
Representation zeta functions of compact p-adic analytic groups and arithmetic groups (2010, background): “Igusa-type zeta integrals at places with bad reduction have been studied in [15].”
Functional equations for zeta functions of groups and rings (2006, methodology): “The Malcev correspondence and a Kirillov-type theory developed by Howe are used to “linearise” the problems of counting subgroups and representations in T-groups, respectively.”
Zeta functions of groups and rings (2007, methodology): “Using various methods from analytic number theory and arithmetic geometry we show in [13] that Proposition 3.”