# What are the algebraic structures of the wallpaper groups?

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?

• Isn't the list/classification of the wallpaper groups precisely what you ask for? – Hagen von Eitzen Nov 2 '15 at 21:38
• I've seen numerous papers that list/classify the wallpaper groups, but none of them seem to give the abstract structures. – Joshua Meyers Nov 2 '15 at 21:52
• Presentations for the wallpaper groups are given in Coxeter and Moser, Generators and Relations for Discrete Groups, Chapter 4 and Table 3. – Chappers Apr 26 '16 at 21:11

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 $$\langle x,y,u,v \mid [x,y],u^2 = x,v^2 = y,xv = x^{−1},yu = y^{−1},(uv)^2 \rangle$$
• Can you describe the structures of the 17 $\Gamma$ explicitly then? – Joshua Meyers Nov 2 '15 at 21:55
• Yes, one can describe this explicitly for all $17$ groups, see chapter $2$ of Ante's thesis. – Dietrich Burde Nov 2 '15 at 22:01