Consider the following situation: $\Gamma\_0\leq\Gamma$ are both finitely generated groups and $\Gamma\_0$ has finite index in $\Gamma$. The restriction gives a well defined map between the character varieties of these groups:
$$\mathrm{Res}:M(\Gamma,\mathrm{GL}\_N(\mathbb{C}))\longrightarrow M(\Gamma\_0,\mathrm{GL}\_N(\mathbb{C}))$$
where $M(\Gamma,\mathrm{GL}\_N(\mathbb{C}))$ is the GIT quotient of the variety $\mathrm{Hom}(\Gamma,\mathrm{GL}\_N(\mathbb{C}))$ under the action of $\mathrm{GL}\_N(\mathbb{C})$ by conjugation. Is it always true that the map $\mathrm{Res}$ is proper ? If not, can you provide a (simple) counterexample ?


Tanks in advance. Benoît



Character varieties of finitely generated groups
Claim. The restriction map is always proper, where the target group $G$ is the group of $K$-points of a reductive group over a local field $K$, e.g. $G=GL\_N({\mathbb C})$.


Proof. First, some generalities, details for which you can find, for instance, [here](http://arxiv.org/abs/1003.1111). Let $X$ be the symmetric space or a locally compact Euclidean building corresponding to $G$. For each representation $\rho$ to $G$ of a group $\Lambda$ with generators $\gamma\_1,...,\gamma\_k$, define the min-max displacement
$$
d\_\rho:=\inf\_{x\in X} d\_\rho(x), d\_\rho(x):=\max\_j \rho(\gamma\_j)(x).
$$ 
Let $x\_\rho\in X$ denote a point for which $d\_\rho(x)-d\_\rho\le 1$. It is not hard to check that the number $d\_\rho$ depends only on the projection $[\rho]$ of $\rho$ to $M(\Lambda, G)$. Furthermore, a sequence $[\rho\_i]$ is precompact in $M(\Lambda, G)$ iff the sequence $(d\_{\rho\_i})$ is bounded. Suppose that $[\rho\_i]$ is not precompact and the sequence $(d\_{\rho\_i})$ diverges to infinity. Then you take the asymptotic cone $Cone(X)$ of $X$ centered at the points $x\_{\rho\_i}$ with scaling factors $(d\_{\rho\_i})^{-1}$. The result is an isometric action $\rho$ of $\Lambda$ on $Cone(X)$ without a common fixed point. The key fact is that $Cone(X)$ is a Euclidean building, by a theorem of Kleiner and Leeb. By the Cartan-Tits theorem, the action $\rho$ has no bounded orbits. 


Now, we can prove the claim. Suppose to the contrary, that there exists a sequence of representations $\rho\_i$ of $\Gamma$ whose projections to $M=M(\Gamma, G)$ diverge, while their restrictions $\rho'\_i$ to $\Gamma\_0$ project to a relatively compact sequence in $M(\Gamma\_0,G)$. Then the above construction yields an action $\rho$ of $\Gamma$ on $Cone(X)$.


Note that the restricted actions $\rho\_i'$ have bounded $d\_{\rho'\_i}$. There are two cases to consider:


1. The distance between points $x\_{\rho\_i'}$ and $x\_{\rho\_i}$ is $O(d\_{\rho\_i})$. Then 
the sequence $(x\_{\rho\_i'})$ represents a point $x$ in $Cone(X)$. This point is necessarily fixed by $\Gamma\_0$. Thus, since $|\Gamma:\Gamma\_0|<\infty$, the $\Gamma$-orbit of $x$ is bounded, which is a contradiction.
2. $d\_{\rho\_i}=o(d(x\_{\rho\_i'},x\_{\rho\_i}))$. Then geodesic segments between $x\_{\rho\_i'},x\_{\rho\_i}$ represent a geodesic ray in $Cone(X)$. The point at infinity $\xi$ represented by this ray is fixed by $\Gamma\_0$ and, moreover, every element of $\Gamma\_0$ acts as a unipotent isometry of $Cone(X)$. Hence, $\Gamma\_0$ has a common fixed point in $X$. Now, the argument is the same as in Case 1. QED.


