Superrigidity theorems in positive characteristic
Let \(A\) be a nonempty finite set. For each \(\alpha\in A\), choose a local field \(k_\alpha\) and a connected simply connected semisimple \(k_\alpha\)-group \(G_\alpha\) without \(k_\alpha\)-anisotropic factors. Denote by \(G_\alpha(k_\alpha)\) the groups of \(k_\alpha\)-rational points of \(G_\alpha\) and let \(G= \prod_{\alpha\in \Lambda} G_\alpha(k_\alpha)\). Let \(\text{rank} G= \sum_{\alpha\in \Lambda} \text{rank }k_\alpha G_\alpha\) denote the rank of the group \(G\). Let \(\Gamma\) be a lattice in \(G\) and \(\Lambda\) be a subgroup of the commensurability group \(\text{comm}_G(\Gamma)\) containing \(\Gamma\). Let \(F\) be an algebraic group defined over a local field \(k\) of positive characteristic such that the group \(H= F/R_u(F)\) is semisimple and \(\tau: \Lambda\to F(k)\) be a homomorphism with \(\tau(\Lambda)\) Zariski dense in \(F\). Suppose the following conditions are satisfied: (i) \(\forall\alpha\in A\), the group \(G_\alpha\) is simply connected. (ii) Either \(\text{rank }G\geq 2\) and \(\Gamma\) is irreducible or the subgroup \(\Lambda\) is dense in \(G\) and each irreducible component of \(\Gamma\) is contained in a subgroup of \(G\) of \(\text{rank}\geq 2\). (iii) The restriction of the homomorphism \(\pi o\tau: \Lambda\to H(k)\) to \(\Gamma\) almost extends to a continuous homomorphism \(\rho: G\to H(k)\) such that \(\rho(G)\neq e\). (iv) The representation \(Ado \tau: \Lambda\to \text{GL}(\text{Dist }F)\) is \(\Gamma\)-integrable. It is proved here that there exist a continuous homomorphism \(\widetilde\tau: G\to F(k)\) and a homomorphism \(\nu:\Lambda\to F(k)\) with \(\nu(\Gamma)\) compact such that \(\widetilde\tau(G)\) and \(\nu(A)\) commute and \(\tau(\lambda)= \nu(\lambda) \widetilde\tau(\Lambda)\) for all \(\lambda\in\Lambda\). This generalizes the superrigidity theorem of Margulis (1991) to the case of the more general non-reductive target group \(F\). The proof follows the general plan of Margulis' proof of his superrigidity theorems. A proof of the cocycle superrigidity theorem in the case of algebraic groups defined over a local field of positive characteristic and generalization of this proof to the case of more general target groups analogous to the generalization of the superrigidity theorem is also given.
scientific article; zbMATH DE number 1496016
scientific article