# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Representing the automorphism group of an almost crystallographic groupHTML articles powered by AMS MathViewer

by Paul Igodt and Wim Malfait
Proc. Amer. Math. Soc. 124 (1996), 331-340 Request permission

## Abstract:

Let $E$ be an almost crystallographic (AC-) group, corresponding to the simply connected, connected, nilpotent Lie group $L$ and with holonomy group $F$. If $L^F = \{1\}$, there is a faithful representation $\operatorname {Aut}{E} \hookrightarrow \operatorname {Aff}(L)$. In case $E$ is crystallographic, this condition $L^F =\{1\}$ is known to be equivalent to $Z(E)=1$ or $b_1(E)=0$. We will show (Example 2.2) that, for AC-groups $E$, this is no longer valid and should be adapted. A generalised equivalent algebraic (and easier to verify) condition is presented (Theorem 2.3). Corresponding to an AC-group $E$ and by factoring out subsequent centers we construct a series of AC-groups, which becomes constant after a finite number of terms. Under suitable conditions, this opens a way to represent $\operatorname {Aut}{E}$ faithfully in $\mathrm {Gl}(k,\mathbb {Z}{}{}) \times \operatorname {Aff}(L_1)$ (Theorem 4.1). We show how this can be used to calculate $\operatorname {Out}{E}$. This is of importance, especially, when $E$ is almost Bieberbach and, hence, $\operatorname {Out}{E}$ is known to have an interesting geometric meaning.
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