Superrigidity theorems in positive characteristic
type: Item
instance of: scholarly article
title: Superrigidity theorems in positive characteristic
zbMATH Open document ID: 0959.22006
author: L. Lifschitz
published in: Journal of Algebra
publication date: 2000-11-16T00:00:00Z
review text: 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 theor…
reviewed by: Ranjeet Sehmi
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 1496016
zbMATH Keywords: lattice
zbMATH Keywords: superrigidity theorem
zbMATH Keywords: target group
zbMATH Keywords: local field of positive characteristic
MaRDI profile type: Publication
full work available at URL: jabr.2000.8315
OpenAlex ID: W2075868920
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DOI: 10.1006/JABR.2000.8315
description: scientific article; zbMATH DE number 1496016