Zeta functions of groups and rings

Abstract

The representation growth of a T -group is polynomial. We study the rate of polynomial growth and the spectrum of possible growth, showing that any rational number ? can be realized as the rate of polynomial growth of a class 2 nilpotent T -group. This is in stark contrast to the related subject of subgroup growth of T -groups where it has been shown that the set of possible growth rates is discrete in Q. We derive a formula for almost all of the local representation zeta functions of a T2-group with centre of Hirsch length 2. A consequence of this formula shows that the representation zeta function of such a group is finitely uniform. In contrast, we explicitly derive the representation zeta function of a specific T2-group with centre of Hirsch length 3 whose representation zeta function is not finitely uniform. We give formulae, in terms of Igusa's local zeta function, for the subring, left-, right- and two-sided ideal zeta function of a 2-dimensional ring. We use these formulae to compute a number of examples. In particular, we compute the subring zeta function of the ring of ?integers in a quadratic number field.

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