Diffuse property of low dimensional Bieberbach groups in CSV format and in a SQL like table.
The above files contain only the results. The gzipped gap file contains more details. The data in the file should be read as follows. Assume that \(\Gamma\) is a Bieberbach group of dimension \(n, n\leq 6\) that fits into the following short exact sequence \[ 0 \longrightarrow M \longrightarrow \Gamma \longrightarrow G \longrightarrow 1, \] where \(M=\mathbb{Z}^n\) is the maximal abelian normal subgroup of \(\Gamma\) and \(G\) is the holonomy group of \(\Gamma\). If the rank of the center of \(\Gamma\) is equal to \(k\), then we have an epimorphism \[ f \colon \Gamma \to \mathbb{Z}^k. \] Note that in this case \(\Gamma \subset \operatorname{GL}(n,\mathbb{Z}) \ltimes \mathbb{Q}^n \subset \operatorname{GL}(n+1,\mathbb{Q})\), up to conjugation in \(\operatorname{GL}(n+1,\mathbb{Q})\), contains of matrices of the form \[ \begin{bmatrix} * & * & *\\ 0 & I_k & *\\ 0 & 0 & 1\end{bmatrix} \] and the following map from \(\ker f\) to \(\operatorname{GL}(n-k+1,\mathbb{Q})\) \[ \begin{bmatrix} A & * & a\\ 0 & I_k & 0\\ 0 & 0 & 1\end{bmatrix} \mapsto \begin{bmatrix} A & a\\0 & 1\end{bmatrix} \] is an isomorphism.
Description of fields of GAP record is given in the following table.
| Field | Description |
|---|---|
| name | CARAT name of the group |
| dim | dimension \(n\) of \(\Gamma\) |
| cgens | generators of the group (without lattice generators) |
| holonomy | GAP id of \(G\) |
| hdiff | is every Sylow subgroup of \(G\) cyclic (and hence \(\Gamma\) is diffuse group) |
| zrank | rank of the center of \(\Gamma\) |
| sub.complement | basis of \(M^G \cong Z(\Gamma)\) |
| sub.sub | complement of sub.complement to a basis of \(M\) |
| kgens | generators of \(\operatorname{ker}f\) as \(n-k\)-dimensional Bieberbach group (without lattice generators) |
| kname | CARAT name of \(\operatorname{ker}f\) |
| diffuse | information about diffuse property of \(\Gamma\) |