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The wallpaper groups are discrete groups of affine motions in the plane that contain two linearly independent translations. Cf. https://en.wikipedia.org/wiki/Wallpaper_group

Some of them have very straightforward structures, like $$\textrm{pm}=\textrm{Dih}_2\times\mathbb{Z}$$ $$\textrm{pg}=\mathbb{Z}\rtimes\mathbb{Z}\quad\textrm{(non-trivial semidirect product)}$$

But the structure of $\textrm{p2gg}$, which is generated by two orthogonal glide reflections, is not straightforward at all. What is the structure of this group? Is there a general reference which contains the structures of all the wallpaper groups?

Joshua Meyers
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1 Answers1

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By the Bieberbach theorems, such a group $\Gamma$, being a discrete cocompact subgroup of $$Isom(\mathbb{R}^2)=O_2(\mathbb{R})\rtimes T(2),$$ has a normal subgroup $N\simeq \mathbb{Z}^2$ of finite index, consisting of translations. Its point group $F=\Gamma/N$ is finite, and is one of the $10$ groups $C_1,C_2,C_3,C_4,C_6,D_1,D_2,D_3,D_4,D_6$. $\Gamma$ is then an extension of $F$ by $N$, i.e., we have the short exact sequence $$ 1\rightarrow \mathbb{Z}^2\rightarrow \Gamma\rightarrow F\rightarrow 1. $$ There are exactly $17$ such different $\Gamma$, and their structure can be described explicitly, e.g., by giving explicit generators and relations for these groups, see the thesis of Antje Meiser, which gives in addition the zeta functions for all $17$ groups.

Example. The presentation for $p2gg$ is given by $$\langle x,y,u,v \mid [x,y],u^2 = x,v^2 = y,xv = x^{−1},yu = y^{−1},(uv)^2 \rangle$$

Dietrich Burde
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