On \(SS\)-rigid groups and A. Weil's criterion for local rigidity. I
Let \(\Gamma\) be a finitely generated group. \(\Gamma\) is said to be \(SS\)-rigid if \(\Gamma\) has only finitely many inequivalent completely reducible representations in each dimension. The following condition (*) is necessary for a finitely generated group \(\Gamma\) to be \(SS\)-rigid: (*) for any subgroup \(\Gamma_1\subset\Gamma\) of finite index, the abelianization \(\Gamma_1^{\text{ab}}\) is finite. Main Theorem: Let \(\Gamma\) be a finitely generated group satisfying (*). Each of the following conditions implies that \(\Gamma\) is \(SS\)-rigid: (i) There exists a constant \(c_1\) such that for any \(n\geq 1\) and any irreducible representation \(\rho\colon\Gamma\to\text{GL}_n(\mathbb{C})\), the dimension of the Zariski closure \(\overline{\rho(\Gamma)}\) is \(\leq c_1\); (ii) There exists a constant \(c_2\) such that for any \(n\geq 1\) and any irreducible representation \(\rho\colon\Gamma\to\text{GL}_n(\mathbb{C})\), \(\dim H^1(\Gamma,\text{Ad}\circ\rho)\leq c_2\). Then the author points out a connection between the condition (i) of the main theorem and the same author's previous results on the \(SS\)-rigidity of groups having bounded generation and satisfying (*). Finally, condition (i) can be verified for \(\text{SL}_n(\mathbb{Z})\), \(n\geq 3\), and \(\text{SL}_2(\mathbb{Z}[1/p])\) by means of a straightforward combinatorial calculation.
scientific article; zbMATH DE number 1239595