Representation varieties of arithmetic groups and polynomial periodicity of Betti numbers (Q1343845)

The paper contains several results about character varieties of arithmetic groups and their congruence representations. An important application is a periodicity theorem for Betti numbers which we formulate below. Let \(\mathcal H\) be an algebraic \(\mathbb{Q}\)-subgroup of \(\text{SL}_m\) and \(\mathcal N\) its unipotent radical. It is supposed that \({\mathcal H}/{\mathcal N}\) is a connected, algebraically simply connected, semisimple \(\mathbb{Q}\)-group with only \(\mathbf R\)-isotropic almost \(\mathbb{Q}\)-simple factors. Denote \(H ={\mathcal H}(\mathbb{Z})\) and, for any integer \(K > 0\), define the congruence subgroup \({\mathcal H}(K)=\{X\in H,X\equiv I\bmod K\}\). Let \(H_K=H/{\mathcal H}(K)\). Let \(M\) be a connected topological space with the homotopy type of a finite CW complex and fix a surjective homomorphism \(\pi_1(M)\to H\). Given a finite dimensional complex representation \(\tau\) of \(H_K\), we obtain a representation of \(\pi_1(M)\) and hence a locally constant sheaf \(V_\tau\) on \(M\). Denote \(\beta^q(M,\tau)=\dim H^q(M,V_\tau)\) and \(\beta^{q,\nu}_K=\nu \sum\beta^q(M,\tau)\), where the summation runs over all irreducible \(\tau\) of dimension \(\nu\). If \(\mathcal H\) has a \(\mathbb{Q}\)-Levi factor \(\mathcal G\) such that \({\mathcal H}(\mathbb{Z})={\mathcal G}(\mathbb{Z}){\mathcal N}(\mathbb{Z})\), then the sequence \(\beta^{q,\nu}_K\), \(K=1,2,\dots,\) is polynomial-periodic (i.e. a polynomial in \(K\) whose coefficients depend on \(K\) periodically). This result generalizes those of \textit{P. Sarnak} and the author [ibid. 88, 31-72 (1994; Zbl 0843.11027)].