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

TL;DR

This paper explicitly computes the lower algebraic K-theory for all 73 split three-dimensional crystallographic groups, providing a general splitting formula and detailed classifications, thereby advancing understanding of their algebraic and geometric structures.

Contribution

It introduces a comprehensive calculation of the lower algebraic K-theory for split 3D crystallographic groups and offers a general splitting formula applicable to all such groups.

Findings

Explicit descriptions of all 73 split groups

A general splitting formula for K-theory

Classification of split crystallographic groups

Abstract

We explicitly compute the lower algebraic K-theory of the split three-dimensional crystallographic groups; i.e., the groups G that act properly and cocompactly on three-dimensional Euclidean space by isometries, such that the natural map from G to O(3) is a split injection onto its image. There are 73 split three-dimensional crystallographic groups in all, out of a total of 219 isomorphism types of three-dimensional crystallographic groups. We also provide a general splitting formula for the lower algebraic K-theory that is valid for all three-dimensional crystallographic groups. This result generalizes earlier work of Alves and Ontaneda. Along the way, we give explicit descriptions of all 73 split three-dimensional crystallographic groups, and completely work out their classification. The split crystallographic groups are sometimes called "splitting groups". A theorem of crystallographic groups says that any crystallographic group is a finite-index subgroup of its splitting group, so each three-dimensional crystallographic group is a finite-index subgroup of one from our list.