The Lower Algebraic K-Theory of Split Three-Dimensional Crystallographic Groups

Abstract

We compute K −1 , K 0 , and the Whitehead groups of all threedimensional crystallographic groups Γ which fit into a split extensionwhere H is a finite group acting effectively on Z 3 . Such groups Γ account for 73 isomorphism types of three-dimensional crystallographic groups, out of 219 types in all.We also prove a general splitting formula for the lower algebraic K-theory of all three-dimensional crystallographic groups, which generalizes the one for the Whitehead group obtained by Alves and Ontaneda.