If you like this topic, I strongly recommend you read the references I attached to my comment. In uniformly curved 2D hyperbolic spaces, it turns out that ther

If you like this topic, I strongly recommend you read the references I attached to my comment. In uniformly curved 2D hyperbolic spaces, it turns out that there is a higher dimensional non-Abelian Fuchsian translation group defined on a higher genus torus.