How many examples exist of Lie groups that are 2-dimesional surfaces?

Question

It is relatively easy to show that $\mathbb{R}^2$ or $\mathbb{T}^2$ are 2-dimensional surfaces with a structure of Lie groups. I can not find other surface which are also a Lie group, there are more examples?

Many thanks!

Answer 1

There's also a cylinder. That's it. You can prove this by fully classifying 2-dimensional Lie groups. It's much easier to classify 2-dimensional Lie algebras, of which there are two up to isomorphism, and hence 2 simply connected 2-dimensional Lie groups up to isomorphism: $\Bbb R^2$ and $\text{Aff}(1)$, the affine transformations of the line. Now one classifies their discrete closed normal subgroups. For $\Bbb R^2$, there are lattices, either isomorphic to $\Bbb Z$ or $\Bbb Z^2$, and the quotient is either a cylinder or a torus, depending. The only normal subgroups of $\text{Aff}(1)$ are translation groups, giving the same result.