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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Lower bounds for dimensions of
representation varieties

Author: Andy R. Magid
Journal: Trans. Amer. Math. Soc. 350 (1998), 4609-4621
MSC (1991): Primary 20C15
MathSciNet review: 1433124
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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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Additional Information

Andy R. Magid
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019

Keywords: Finitely generated roups, linear representations, varieties, cohomolgy
Received by editor(s): May 25, 1995
Received by editor(s) in revised form: November 25, 1996
Additional Notes: Partially supported by NSA grant MDA904–92–H–3038
Article copyright: © Copyright 1998 American Mathematical Society