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It is well known that geometrically finite Fuchsian groups, or finitely generated discrete subgroups of $\textrm{PSL}_2(\mathbb{R})$ can be classified up to isomorphism by their signature $[g,s;m_1,\dots,m_r]$ which define a presentation

$\Gamma = \left\langle a_1,b_1,\dots,a_g,b_g,c_1\dots,c_s,d_1,\dots,d_r\ |\ d_1^{m_1}=\dots=d_r^{m_r}=\prod_{i=1}^g[a_i,b_i]\prod_{j=1}^{r}d_j\prod_{k=1}^sc_k=1 \right\rangle.$

Does there exist a similar presentation for discrete subgroups of $\textrm{PGL}_2(\mathbb{R})$?

Sam
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  • What do you know about orbifolds? For instance, do you know how to derive that presentation from the form of the quotient orbifold $\mathbb H^2 / \Gamma$? – Lee Mosher Jul 14 '19 at 20:09
  • I'm familiar with Poincaré's polygon theorem for fundamental domains of Fuchsian groups. However, I'm not sure how to generalise the version I've seen to other quotient spaces. – Sam Jul 14 '19 at 21:39
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    You may want to read Macbeath's paper "The Classification of Non-Euclidean Plane Crystallographic Groups"; it contains presentations of NEC groups and much more; https://www.cambridge.org/core/services/aop-cambridge-core/content/view/9AB398CB5C92F95433B1801807512AF2/S0008414X00055164a.pdf/classification_of_noneuclidean_plane_crystallographic_groups.pdf – Moishe Kohan Jul 15 '19 at 04:42
  • @MoisheKohan that is excellent and definitely covers what I'm looking for – Sam Jul 15 '19 at 13:00
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    @MoisheKohan: That would be a good thing to post as an answer. I searched a while on the site here to see if there was already an appropriate answer, and found none. – Lee Mosher Jul 15 '19 at 13:14
  • @YCor I was being too hasty with the question, edited. – Sam Jul 16 '19 at 20:37
  • Is the difference in the non-orientable case, that $a_1,\dots,b_g$ is replaced with $u_1,\dots, u_g$ and $\prod [a_i,b_i]$ is replaced with $\prod u_i^2$? – YCor Jul 16 '19 at 20:41

1 Answers1

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Geometrically speaking, $PGL(2,{\mathbb R})$ is the full isometry group of the hyperbolic plane ${\mathbb H}^2$. NEC (Noneuclidean crystallographic) groups are discrete isometry groups of ${\mathbb H}^2$ acting cocompactly.

You may want to read Macbeath's paper "The Classification of Non-Euclidean Plane Crystallographic Groups"; it contains presentations of NEC groups and much more. The issue, however, is that the paper deals only NEC groups. One can write down a general presentation for geometrically finite isometry groups of the hyperbolic plane by taking Macbeath's presentations and declaring some of the exponents $m_i$ to be zero. (There will be a few exceptional cases that will be missing: These are finite extensions of cyclic groups.) The proof that this is a complete list of presentations will be essentially the same as in Macbeath's paper.

Moishe Kohan
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