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I was thinking of Veech surfaces, which are translation surfaces whose stabilizer under the $\mathrm{Sl}_2(\mathbb{R})$ action is a lattice in $\mathrm{Sl}_2(\mathbb{R})$. They seem to have been studied and examples of such surfaces are rare and somehow well understood in low genus.

I was wondering what other examples of Veech groups were known? For example do we know explicit examples of translation surfaces whose Veech group is non-elementary but not a lattice?

Thank you for your attention!

Selim

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Infinitely generated Veech groups were discovered by Curtis McMullen and Pascal Hubert and Thomas Schmidt. Both results are more-or-less explicit (in describing a specific translation surface whose Veech group is infinitely generated). Hubert and his collaborators have many other results on Veech groups.

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