Representing the automorphism group

of an almost crystallographic group

Authors:
Paul Igodt and Wim Malfait

Journal:
Proc. Amer. Math. Soc. **124** (1996), 331-340

MSC (1991):
Primary 20H15, 20F34, 20F28

DOI:
https://doi.org/10.1090/S0002-9939-96-03141-3

MathSciNet review:
1301030

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an almost crystallographic (AC-) group, corresponding to the simply connected, connected, nilpotent Lie group and with holonomy group . If , there is a faithful representation . In case is crystallographic, this condition is known to be equivalent to or . We will show (Example 2.2) that, for AC-groups , this is no longer valid and should be adapted.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 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 faithfully in (Theorem 4.1). We show how this can be used to calculate . This is of importance, especially, when is almost Bieberbach and, hence, is known to have an interesting geometric meaning.

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Additional Information

**Paul Igodt**

Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven Campus Kortrijk, Universitaire Campus, B-8500 Kortrijk, Belgium

**Wim Malfait**

Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven Campus Kortrijk, Universitaire Campus, B-8500 Kortrijk, Belgium

DOI:
https://doi.org/10.1090/S0002-9939-96-03141-3

Keywords:
Almost crystallographic group,
automorphism group,
outer automorphism group

Received by editor(s):
May 5, 1994

Additional Notes:
The second author is Research Assistant of the National Fund For Scientific Research (Belgium)

Communicated by:
Ron Solomon

Article copyright:
© Copyright 1996
American Mathematical Society