Representing the automorphism group of an almost crystallographic group

Igodt, Paul; Malfait, Wim

doi:10.1090/S0002-9939-96-03141-3

Representing the automorphism group of an almost crystallographic group
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by Paul Igodt and Wim Malfait PDF

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 We will show (Example 2.2) that, for AC-groups $E$, this is no longer valid and should be adapted.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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