Cohomologies of regular lattices over the Kleinian 4-group

TL;DR

This paper explicitly computes the cohomologies of lattices over the Kleinian 4-group within regular components, applying the results to classify certain crystallographic and Chernikov groups.

Contribution

It provides explicit cohomology calculations for lattices over the Kleinian 4-group, a novel contribution to the understanding of their structure and applications.

Findings

Explicit cohomology formulas for lattices over the Kleinian 4-group

Classification of some crystallographic groups

Classification of Chernikov groups

Abstract

We calculate explicitly cohomologies of the lattices over the Kleinian 4-group belonging to the regular components of the Auslander-Reiten quiver as well as of their dual modules. The result is applied to the classification of some crystallographic and Chernikov groups.