Experimental Mathematics

On Symmetry of Flat Manifolds

Rafał Lutowski

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Abstract

We give an example of a Bieberbach group $\Gamma$ for which $\Out(\Gamma)$ is a cyclic group of order $3$. We also calculate the outer automorphism group of a direct product of $n$ copies of a Bieberbach group with trivial center, for $n \in \mathbb{N}$. As a corollary we get that every symmetric group can be realized as an outer automorphism group of some Bieberbach group.

Article information

Source
Experiment. Math., Volume 18, Issue 2 (2009), 201-204.

Dates
First available in Project Euclid: 25 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.em/1259158430

Mathematical Reviews number (MathSciNet)
MR2549689

Zentralblatt MATH identifier
1202.20052

Subjects
Primary: 20H15: Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25]
Secondary: 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx] 57S30: Discontinuous groups of transformations

Keywords
Bieberbach groups directly indecomposable groups

Citation

Lutowski, Rafał. On Symmetry of Flat Manifolds. Experiment. Math. 18 (2009), no. 2, 201--204. https://projecteuclid.org/euclid.em/1259158430


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