# Diffuse property of low dimensional Bieberbach groups

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.

FieldDescription
nameCARAT name of the group
dimdimension $$n$$ of $$\Gamma$$
cgensgenerators of the group (without lattice generators)
holonomyGAP id of $$G$$
hdiffis every Sylow subgroup of $$G$$ cyclic (and hence $$\Gamma$$ is diffuse group)
zrankrank of the center of $$\Gamma$$
sub.complementbasis of $$M^G \cong Z(\Gamma)$$
sub.subcomplement of sub.complement to a basis of $$M$$
kgensgenerators of $$\operatorname{ker}f$$ as $$n-k$$-dimensional Bieberbach group (without lattice generators)
knameCARAT name of $$\operatorname{ker}f$$
diffuseinformation about diffuse property of $$\Gamma$$