Let *K* be a group (with discrete topology), *G* be a Lie group. Let $\operatorname{Hom}(K,G)$ be the space of all group homomorphisms from *K* to *G*. This is a closed subvariety of the space of all the maps from the generators of *K* to *G*, and as such has a topology.

Andre Weil's paper "On Discrete Subgroups of Lie Groups" proves that an important subset $U \subset \operatorname{Hom}(K,G)$ is open. *U* is defined as the set of all homomorphism $K\to G$ such that the homomorphism is injective, the image is discrete, and the quotient $G/image(K)$ is compact.

## Questions:

What happens if you remove the condition that the quotient is compact?

How often/where is this taught? What kinds of books would it be in, what kind of courses would have it? This looks like a basic result that could be taught anywhere, but it's completely new to me (not that I know much about representation theory). while Weil's paper fortunately seems very readable, I couldn't easily find any other source that would such questions.

## Motivation:

In the case where $K=\pi_1 (S)$ is the fundamental group of a surface and $G=PSL_2(\mathbb R)$, the space $\operatorname{Hom}(K,G)$ is very closely related to the Teichmuller space of *S*. Every Riemann surface is a quotient of $PSL_2(\mathbb R)$ by a discrete subgroup. So, for an element of $\operatorname{Hom}(K,G)$, the quotient $G/image(K)$ corresponds to a Riemann surface and the data of the actual map $K\to G$ gives a marking on it.

Not every homomorphism $K\to G$ corresponds to a point of Teichmuller space. For example, the map that sends all of *K* to the identity is clearly no good, as the quotient $K/G$ is not topologically the same as the surface *S*. However, if the map is injective and the image of *K* in *G* is discrete, all will be well. So, Weil's theorem basically says that the Teichmuller space of *S* is an open subset of $\operatorname{Hom}(\pi_1(S),PSL_2(\mathbb R))$.

However, since Weil's theorem requires the quotient to be compact, this won't work if *S* is a *non-compact* Riemann surface. I wonder how much more difficult life becomes in this case.

## Disclaimer/Another question:

The above has a small lie in it. To get the Teichmuller space, you actually need to look at the quotient $\operatorname{Hom}(K,G)/G$ where *G* acts on $\operatorname{Hom}(K,G)$ by conjugation of the target. In the case of compact surfaces, this is not supposed to mess up the fact that the subset is open; this seems to be a result of William Goldman but I don't have the exact reference. If you can say anything about this, I'd appreciate it too.

Thank you very much!