Computations with almost-crystallographic groups

Summary

Abstract

Recently, algorithmic approaches to construct and investigate almost crystallographic groups and a library of almost crystallographic groups of small Hirsch length have been made available in the Aclib package of Gap. Here we present a survey of these methods and we illustrate a variety of their applications.

Introduction

Almost crystallographic groups have first been discussed in the theory of actions on connected and simply connected nilpotent Lie groups L. In this setting L ⋊ Aut(L) acts affinely on L via l^(m,α) = l^α · m for l,m ∈ L and α ∈ Aut(L). If C is a maximal compact subgroup of Aut(L), then a subgroup G of L⋊C is almost crystallographic if the action of G on L is properly discontinuous and the quotient space L/G is compact. Almost crystallographic groups can also be characterized as those finitely generated nilpotent-by-finite groups whose normal torsion subgroup is trivial. One of the most fundamental observations on almost crystallographic groups is that for a given finitely generated torsion-free nilpotent group N there exist only finitely many almost crystallographic groups having N as Fitting subgroup. This property can be used as a basis for a classification of almost crystallographic groups. In fact, in [2] this approach has been exploited to determine a library of almost crystallographic groups of Hirsch length at most 4. Recently, this library of almost crystallographic groups has been made available in electronic form in the package Aclib [3] of the computer algebra system Gap [20].