Virtually abelian symmetry groups of hyperbolic lattices

TL;DR

This paper classifies integral lattices with virtually abelian symmetry groups and completes the classification of K3 surfaces with such automorphism groups, also providing an algorithm for lattice isometry testing.

Contribution

It introduces a classification of lattices with virtually abelian symmetries and completes the classification of related K3 surfaces, along with an algorithm for lattice isometry determination.

Findings

Classification of integral lattices with virtually abelian symmetry groups

Completion of K3 surface automorphism group classification

Algorithm for weak approximation in orthogonal groups

Abstract

We give a classification of integral lattices with virtually abelian symmetry group. As a consequence, we complete the classification of K3 surfaces with virtually abelian automorphism group. In the appendix we formulate an algorithm for weak approximation in orthogonal groups and use it to determine if two indefinite lattices are isometric.