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The only Fuchsian groups I know of (apart from silly ones like finite cyclic groups) are the subgroups of triangle groups. For instance, the modular group $\text{PSL}(2,\mathbb{Z})$.

I've only ever seen hard-to-construct examples of others.

  1. Can anyone give some easier-to-construct examples?
  2. Is there some sort of classification of Fuchsian groups, or of a large subclass of them?

There must exist a lot since, as the link says, almost all Fuchsian groups are non-triangular.

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There are plenty of easy to construct examples of Arithmetic Fuchsian Groups; see also the book of Svetlana Katok on Fuchsian groups. All arithmetic triangle Fuchsian groups are classified, see here. So it easy to find new examples.

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