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



Non-lattice Veech groups
Infinitely generated Veech groups were discovered by [Curtis McMullen](http://link.springer.com/article/10.1007%2FBF02392964) and [Pascal Hubert and Thomas Schmidt](http://projecteuclid.org/euclid.dmj/1084479318). 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. 


