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ZETA FUNCTIONS OF CRYSTALLOGRAPHIC GROUPS AND ANALYTIC CONTINUATION

Published online by Cambridge University Press:  01 November 1999

M. P. F. DU SAUTOY ,
J. J. McDERMOTT  and
G. C. SMITH
M. P. F. DU SAUTOY
Affiliation:
DPMMS, 16 Mill Lane, Cambridge CB2 1SB, email:[email protected]
J. J. McDERMOTT
Affiliation:
School of Mathematics, Aras na Laoi, University College of Cork, Cork, Ireland, email:[email protected]
G. C. SMITH
Affiliation:
Department of Mathematics, University of Bath, Bath BA2 7AY, email:[email protected]
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Abstract

We prove that the zeta function and normal zeta function of a virtually abelian group have meromorphic continuation to the whole complex plane. We do this by relating the functions to classical L-functions of arithmetic orders considered by Hey, Solomon, Bushnell and Reiner. We calculate the zeta functions and normal zeta functions of the plane crystallographic groups. As a corollary of these calculations we produce \begin{enumerate} \item[(1)] examples of two isospectral residually finite groups with non-isomorphic profinite completions and even distinct lattices of subgroups; and \item[(2)] examples of non-nilpotent residually finite groups whose zeta functions enjoy an Euler product.

1991 Mathematics Subject Classification: 11M41, 20H15.

Keywords

zeta functionanalytic continuationcrystallographic groups
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 79 , Issue 3 , November 1999 , pp. 511 - 534
Copyright
1999 London Mathematical Society

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