# Wallpaper groups for the hyperbolic plane

I would be grateful if someone could direct me to a reference that classifies the equivalent of the wallpaper groups (and the frieze groups and the point groups, if possible) for the hyperbolic plane, i.e., the symmetry groups of isometries acting properly discontinuously on the hyperbolic plane.

The Euclidean case is classical, well-known and easy to find proofs for. The spherical case is also well-known, and normally described in the same terms (see, for example, the Wikipedia article List of spherical symmetry groups). But it seems rather more difficult to find a place where the same ideas for the hyperbolic plane are described in the same way (although I'm sure this has been done), as rotations, translations, reflections, and possibly glide reflections, acting on some sort of fundamental region, as seen in Wikipedia's article on the wallpaper groups, for example.

There are several ways of approaching hyperbolic symmetries, including groups of reflections in the sides of $(l,m,n)$ hyperbolic triangles with $1/l+1/m+1/n<1$, (see e.g. Wikipedia's article), or tessellations of $p$-sided polygons, $q$ of which meet at each vertex, with $(p-2)(q-2)>4$; presumably each hyperbolic wallpaper group appears as a symmetry group of one of these tessellations, and is a subgroup of some triangle group; is there a reference that demonstrates this?

• "Discreteness" is not enough to have a reasonable classification. You will want additional hypotheses. Finite generation is a must. Co-compact is very reasonable, co-finite area is also reasonable and is a wider net, and both of these imply finite generation. Finite generation all by itself is also workable, and casts a still wider net. – Lee Mosher Apr 26 '16 at 23:48
• @LeeMosher , thank you for your reply; looks like my terminology's a bit rusty! I think what I probably mean is isometry groups that act properly discontinuously. Is there some pathological example I'm forgetting that means, under these assumptions, f.g. is not implied, and as necessary as you suggest? – Chappers Apr 27 '16 at 11:49
• I also see that co-finite area is an equivalent of the wallpaper groups' defining condition, but one might also wish to include "hyperbolic frieze groups", where this is not so; is there an example, Euclidean or hyperbolic, of such a group that is not f.g.? My intuition's not very good here; I'm just trying to be as general as is sensible. – Chappers Apr 27 '16 at 11:49
• The Riemann Mapping Theorem implies that for every connected, oriented surface $S$ with countable basis, except for the sphere, torus, and annulus, there is a free and properly discontinuous subgroup $\Gamma$ of isometries of $\mathbb{H}^2$ such that $\Gamma$ is isomorphic to $\pi_1 S$ and the quotient space $\mathbb{H}^2/\Gamma$ is homeomorphic to $S$. There are many surfaces with infinitely generated fundamental group to which this applies, three of which are described here: en.wikipedia.org/wiki/Jacob%27s_ladder_surface – Lee Mosher Apr 27 '16 at 15:02

What you are asking for is the theory of hyperbolic 2-dimensional orbifolds. It's a very big theory. If you do not add additional hypotheses, it becomes somewhat unreasonable to expect a good description of the theory. Even if you add enough hypotheses to tame the question, it still requires a lot of mathematics to even describe the classification.

Let me tame the question a bit by adding some hypotheses and putting some restrictions on the classification, just in order to give a taste of the big picture.

First, I'll add the hypothesis that the group $\Gamma$ is finitely generated. Next, I'll add a stronger hypothesis of co-compactness, meaning $\Gamma$ has a compact fundamental domain. Co-finite area is also a good hypothesis, as said in my comment, but is trickier to describe correctly, although not really any deeper mathematically. So I am focussing on discrete, co-compact groups $\Gamma < \text{Isom}(\mathbb{H}^2)$ (which, as a consequence, side-steps any questions about analogues of Frieze groups).

The basic level is classification up to isomorphism: the group $\Gamma$ is determined by its quotient orbifold $M_\Gamma = \text{H}^2 / \Gamma$, in the sense that $\Gamma,\Gamma' < \text{Isom}(\mathbb{H}^2)$ are isomorphic if and only if their quotient hyperbolic orbifolds $M_\Gamma,M_{\Gamma'}$ are "orbifold equivalent", meaning that there is a homeomorphism $M_\Gamma \to M_{\Gamma'}$ which preserves the orbifold structure.

On this basic level, the classification in the hyperbolic case is identical (in theory) to the Euclidean and spherical cases: the groups $\Gamma$ with above hypotheses care classified by the closed hyperbolic orbifolds; the 17 wallpaper groups are classified up to isomorphism by the 17 closed Euclidean orbifolds up to orbifold equivalence; and there is a similar spherical classification.

There are infinitely many distinct closed hyperbolic 2-orbifolds $\mathcal{O}$ up to equivalence, and hence infinitely many groups $\Gamma$ up to isomorphism. It is possible to completely list all closed hyperbolic 2-orbifolds $\mathcal{O}$ by keeping track of certain invariants:

• the topological type of the surface $S$ underlying $\mathcal{O}$ (topological Euler characteristic $\chi(S)$, orientability of $S$, number of boundary components of $S$);
• the numbers of cone points of all angles in the interior of $S$;
• the cycles of dihedral points going around each boundary component of $S$ (keeping track of angles of the dihedral points in the cycle, and taking appropriate care of orientation of boundary components depending on whether $S$ itself is orientable).

Of course one must also be sure to discard from this list the spherical, Euclidean, and bad 2-orbifolds, but that can be done using the sign of the orbifold Euler characteristic $\chi(\mathcal{O})$, which can be computed by a formula using the invariants listed above, with the following outcome:

• spherical or bad is equivalent to $\chi(\mathcal{O})>0$;
• Euclidean is equivalent to $\chi(\mathcal{O})=0$;
• hyperbolic is equivalent to $\chi(\mathcal{O})<0$.

The link provided in the first line of the answer gives a complete list of the Euclidean, spherical, and bad cases; everything else is hyperbolic.

You have asked for references. One reference is anything about Conway's orbifold notation. You might also look up any references on Fuchsian groups, although this will cover only the cases where $\Gamma$ preserves orientation (in particular, no reflections). Also, the literature on Fuchsian groups tends to focus even more strictly on the case where the groups have no finite order elements (no rotations), but still one can find some references without that restriction.