Lower bounds for dimensions of representation varieties

Abstract

The set of $n$–dimensional complex representations of a finitely generated group $\Gamma$ form a complex affine variety $R_{n}(\Gamma )$. Suppose that $\rho$ is such a representation and consider the associated representation $Ad \circ \rho$ on $n \times n$ complex matrices obtained by following $\rho$ with conjugation of matrices. Then it is shown that the dimension of $R_{n}(\Gamma )$ at $\rho$ is at least the difference of the complex dimensions of $Z^{1}(\Gamma , Ad \circ \rho )$ and $H^{2}(\Gamma , Ad \circ \rho )$. It is further shown that in the latter cohomology $\Gamma$ may be replaced by various proalgebraic groups associated to $\Gamma$ and $\rho$.