Zeta Functions of Crystallographic Groups and Analytic Continuation

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 (1)examples of two isospectral residually finite groups with non-isomorphic profinite completions and even distinct lattices of subgroups; and (2)examples of non-nilpotent residually finite groups whose zeta functions enjoy an Euler product. 1991 Mathematics Subject Classification: 11M41, 20H15.