The Auslander conjecture for dimension less then 7

TL;DR

This paper proves Auslander's conjecture for dimensions less than 7, showing all crystallographic subgroups of the affine group are virtually solvable using mainly dynamical methods.

Contribution

It extends the proof of Auslander's conjecture to dimensions under 7, primarily employing dynamical arguments with some cohomological techniques.

Findings

Confirmed Auslander's conjecture for n<7

Demonstrated all such groups are virtually solvable

Used mainly dynamical methods in the proof

Abstract

In 1964 L. Auslander conjectured that every crystallographic subgroup of an the affine group is virtually solvable, i.e. contains a solvable subgroup of finite index. D. Fried and W. Goldman proved Auslander's conjecture for n = 3 using cohomological arguments. We prove the Auslander conjecture for n < 7. The proof is based mainly on dynamical arguments. In some cases, we use the cohomological argument which we could avoid but it would significantly lengthen the proof.