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I am interested in the set of all non-cocompact Fuchsian lattices which all have a distinguished point as cusp, say $\infty$ in the upper half plane model of the hyperbolic plane. Of course, the Fuchsian lattices may have as many cusps as they like as long as $\infty$ is one of them.

Does there exist a classification of the maximal elements of this set?

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  • $\begingroup$ What sort of classification do you expect? $\endgroup$ – Igor Rivin Oct 20 '13 at 22:48
  • $\begingroup$ @Igor Rivin: something of the form "the maximal elements are the Fuchsian lattices from the following list...." or "the maximal elements are the lattices which have a fundamental domain of this and that form" $\endgroup$ – Maik Köster Oct 21 '13 at 6:17
  • $\begingroup$ And by "maximal" you mean "of minimal volume"? $\endgroup$ – Igor Rivin Oct 21 '13 at 13:46
  • $\begingroup$ I consider the lattices to be partially ordered by inclusion. In that sense I mean maximal. The background of my question is as follows: Sometimes there are constructions/properties which are inherited by finite-index subgroups. So if I want to prove this property for all Fuchsian lattices, it suffices to prove it for the (inclusion-)maximal ones. Therefore I wonder whether there is this classification of these maximal Fuchsian lattices (for me only those with cusps are interesting). $\endgroup$ – Maik Köster Oct 21 '13 at 14:41
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I don't think there's a simple description of these. For any cover $X\to E$ of surfaces (or orbifolds, if you allow Fuchsian groups with torsion?), there is a corresponding map between moduli spaces of $E$ to the moduli space of $X$. So I think to determine the maximal groups corresponding to a surface $\Sigma_{g,p}$ with genus $g$ and $p$ punctures, you would have to construct all possible (topological) covers of $\Sigma_{g,p}\to \Delta$, and look at the image of the moduli space of $\Delta$ in the moduli space of $\Sigma_{g,p}$. Then the maximal guys would be the ones that missed the images of these sub-moduli spaces. I believe in fact that the images will be subvarieties of the Deligne-Mumford compactification, so this space should be something like a quasi-projective variety (or something more general like a scheme or stack), but I don't know of surfaces in which this has been explicitly computed. However, I haven't done a literature search, but hopefully this might indicate how to make your question more precise, and search the relevant literature.

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