I am thinking about homomorphisms $\mathrm{Hom}(\Gamma,G)$, where $G$ is a Lie group and $\Gamma$ is a discrete, finitely generated subgroup.


[This question](https://mathoverflow.net/questions/34640/local-vs-infinitesimal-rigidity) talked about the difference of infinitesimal rigidity vs. local rigidity and [the second answer](https://mathoverflow.net/questions/34640/local-vs-infinitesimal-rigidity/34685#34685) shows that infinitesimal rigidity implies local rigidity. Are there examples (preferably in the context of discrete subgroups of Lie groups) such that $\mathrm{Hom}(\Gamma,G)$ is locally rigid, but not infinitesimally rigid?



Infinitesimal rigidity vs. local rigidity
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 representation $\rho\_0$ of a Euclidean Coxeter group of rank 3 (namely, $T(3,3,3)$, the affine Weyl group of type $A\_2$) to $SO(3)$, see their book "Varieties of Representations of Finitely Generated Groups". In general, character schemes of Coxeter groups satisfy "universality" ("Murphy's law"): All singularities defined over the integers are realized as germs of character varieties. For instance, you could take the non-reduced scheme $x^{175}=0$ and realize it as a singularity of a character variety (to $PO(3)$) of an appropriate Coxeter group. The corresponding representation will have infinitesimal deformations of orders $1$ through $174$, but be locally rigid. 


On the other hand, as far as I can tell, there are no current examples of the following: $\Gamma\subset G$ is a discrete subgroup of a Lie group $G$ and $\rho$ is the identity representation, which is locally but not infinitesimally rigid. (This is impossible for discrete f.g. discrete subgroups in $O(3,1)$, however.) It does not mean that such examples do not exist, I am pretty sure they do, just nobody bothered with making computations. 


Here is one possible construction: Let $\Gamma\subset SU(n,1)$, $n\ge 1$, be a uniform lattice and let $\rho: \Gamma\to SU(n+1,1)$ be the natural representation. Suppose that $\Gamma$ has finite abelianization, while $H^1(\Gamma, {\mathbb C}^{n,1})\ne 0$, where $\Gamma$ acts on ${\mathbb C}^{n,1}$ through its embedding in $SU(n,1)$. Then $\rho$ would be locally but not infinitesimally rigid, see the paper of Goldman and Millson ["Local rigidity of discrete groups acting on complex hyperbolic space"](http://www2.math.umd.edu/~wmg//LocalRigidity.pdf). It is well-known how to construct $\Gamma$'s with both finite abelianization and with nonzero $H^1(\Gamma, {\mathbb C}^{n,1})$. However, having both properties simultaneously is tricky. Another possible candidate is to take the Coxeter example of Lubotzky and Magid. Then $\rho\_0: \Gamma\to O(2)$ is the linear part of the action $\rho$ of $\Gamma=T(3,3,3)$ on ${\mathbb R}^2$ as a Euclidean lattice. Now, there is a decent chance that $\rho$ is locally but not infinitesimally rigid as a representation $\Gamma\to G=Isom({\mathbb R}^3)$. 


Update: Here is an example of a locally rigid but not infinitesimally rigid discrete embedding. 
Consider $\Gamma=T(6,6,6)$, the Coxeter group whose coxeter graph is the triangle with three edge-labels 3. Let $\phi: \Gamma\to \Gamma\_0=T(3,3,3)$ be the natural epimorphism to the affine Coxeter group of type $A\_2$, which sends generators to generators. Let $\rho\_0: \Gamma\_0\to O(2)\subset SO(3)$ be the representation as above. Now, let $\rho=\rho\_1\times\rho\_2: \Gamma\to Isom({\mathbb H}^2)\times SO(3)$ be the product homomorphism, where $\rho\_1$ is the discrete embedding of $\Gamma$ to the isometry group of the hyperbolic plane and $\rho\_2=\rho\_0\circ \phi$. Now, each factor of $\rho$ is rigid, hence $\rho$ is also rigid. The representation $\rho$ is a discrete embedding since $\rho\_1$ is. However, $\rho$ is not infinitesimally rigid, since $\rho\_2$ is not (since $\rho\_0$ was not). 


