A formula for the Whitehead group of a three-dimensional crystallographic group

We give an explicit formula for Whitehead group of a threedimensional crystallographic group Γ in terms of the Whitehead groups of the virtually infinite cyclic subgroups of Γ. 0 Introduction The K-theory of integral group rings of crystallographic groups has been studied by Connolly and Koźniewski [8], Farrell and Hsiang [10], Farrell and Jones [11], Lück and Stamm [15], Quinn [20], Tsapogas [25]. Also, in [18], Pearson gives explicit computations of the lower algebraic K-theory of twodimensional crystallographic groups. In this article we give a simple formula for Whitehead group of a threedimensional crystallographic group Γ in terms of the Whitehead groups of the virtually infinite cyclic subgroups of Γ. Here is our main result: Main Theorem: Let Γ be a 3-crystallographic group. Then