Almost-Bieberbach Groups: Affine and Polynomial Structures

Karel Dekimpe (auth.)9783540618997, 3-540-61899-6

Starting from basic knowledge of nilpotent (Lie) groups, an algebraic theory of almost-Bieberbach groups, the fundamental groups of infra-nilmanifolds, is developed. These are a natural generalization of the well known Bieberbach groups and many results about ordinary Bieberbach groups turn out to generalize to the almost-Bieberbach groups. Moreover, using affine representations, explicit cohomology computations can be carried out, or resulting in a classification of the almost-Bieberbach groups in low dimensions. The concept of a polynomial structure, an alternative for the affine structures that sometimes fail, is introduced.

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Table of contents :
Preliminaries and notational conventions….Pages 1-11
Infra-nilmanifolds and Almost-Bieberbach groups….Pages 13-30
Algebraic characterizations of almost-crystallographic groups….Pages 31-46
Canonical type representations….Pages 47-102
The Cohomology of virtually nilpotent groups….Pages 103-120
Infra-nilmanifolds and their topological invariants….Pages 121-157
Classification survey….Pages 159-230