Analytic properties of zeta functions and subgroup growth

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: