Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T20:43:14.035Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)