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Zeta functions of groups and rings

Zeta functions of groups and rings
Zeta functions of groups and rings
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.
Snocken, Robert
961b6254-203e-4578-859a-13ed0317831e
Snocken, Robert
961b6254-203e-4578-859a-13ed0317831e
VOLL, CHRISTOPHER
b7bd2890-38ac-4e05-adb7-3b376916ff79

Snocken, Robert (2014) Zeta functions of groups and rings. University of Southampton, School of Mathematics, Doctoral Thesis, 119pp.

Record type: Thesis (Doctoral)

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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Published date: December 2014
Organisations: University of Southampton, Mathematical Sciences

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Local EPrints ID: 372833
URI: https://eprints.soton.ac.uk/id/eprint/372833
PURE UUID: 68f87323-a1c5-4116-9aaa-12be83b2c04f

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Date deposited: 19 Jan 2015 12:55
Last modified: 17 Jul 2017 21:38

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