Lower bounds for dimensions of representation varieties

Magid, Andy

doi:10.1090/S0002-9947-98-01996-5

Lower bounds for dimensions of representation varieties
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by Andy R. Magid PDF

Trans. Amer. Math. Soc. 350 (1998), 4609-4621 Request permission

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 $\rho$.

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