On the character varieties of finitely generated groups. (Q2347761)

The author establishes three results dealing with the character varieties of finitely generated groups. Let \(\Gamma\) be a finitely generated group. The character variety \(X_n(\Gamma)\) parametrizes conjugacy classes of completely reducible \(n\)-dimensional representations of \(\Gamma\) (in characteristic 0). The first result states that if \(\dim X_n(\Gamma)\) does not vanish for all \(n\), then it grows at least linearly. Let \(R\) be a finitely generated commutative ring. The second result concerns the case where \(\Gamma\) is the elementary subgroup \(G(R)^+\) of the group of \(R\)-rational points of a universal Chevalley-Demazure group scheme \(G\) with irreducible root system \(\Phi\) of rank at least two. Assume \(2\in R^\times\) if \(\Phi\) contains \(B_2\), and \(6\in R^\times\) if \(\Phi=G_2\). The second result then gives an upper estimate \(\dim X_n(\Gamma)\leq c\cdot n\) with \(c\) depending on \(R\) only. Now let \(S\) be an affine algebraic \(\mathbb Q\)-variety. The third result is the existence of a finitely generated group \(\Gamma\) with Kazhdan property (T) and an integer \(n\geq 1\) so that there is a biregular \(\mathbb Q\)-defined isomorphism of complex varieties \(S(\mathbb C)\to X_n(\Gamma)\setminus\{[\rho_0]\}\), where \([\rho_0]\) is the isolated point corresponding with the trivial representation.