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First, there are many examples of representations $\rho: \Gamma \to G$ which are locally but not infinitesimally rigid. The earliest example is due to Lubotzky and Magid, it is a reducible representat …

This fact holds for general Lie groups $G$ (which I will equip with a real-analytic structure) and finitely-generated groups $\Gamma$. It is explained in detail in Raghunathan's book "Discrete subgrou …

I do not know about the general deformation theory (if there is such a thing), so I will talk about the special case I am familiar with, namely, representation varieties $R=Hom(\pi,G)$ of representati …

Here are few well-known examples which are not of algebro-geometric nature, where a problem was solved via a reduction to a deformation problem/moduli space problem: Donaldson's work on intersection …

If you take, say, set of real points of the group-scheme $O(n)$, i.e., $O(n, {\mathbb R})$, then you recover the usual orthogonal (real Lie) group, which you know as $O(n)$. Same applies to $SL(n)$, e …

Here is an alternative take on Donu's argument: Removing the image $\sigma$ of a section of $E_\Gamma$ allows one to regard a fiber of $E_{\Gamma}$ as a once-punctured torus $S$. (In order to constr …