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On superrigidity and arithmeticity of lattices in semisimple groups over local fields of arbitrary characteristic. (English) Zbl 0649.22008
The author proves that if \(\Gamma\) is an irreducible lattice in a group G of the form \(\prod^{n}_{i-1}G_ i(k_ i)\), where \(G_ i\) is an absolutely simple group of adjoint type defined and isotropic over a local field \(k_ i\), with \(char(k_ i)\) arbitrary, and if G’ is an absolutely simple group of adjoint type defined over a local field k, \(\rho: \Gamma\to G'(k)\) is a homomorphism with image Zariski dense on G’ and not relatively compact in \(G'(k)\), then \(\rho\) extends to a continuous homomorphism of G into \(G'(k)\) provided \(\sum^{n}_{i=1}k_ i\)-rank\((G_ i)\geq 2\). As a consequence it is proven that \(\Gamma\) is arithmetic when \(\Gamma\) is finitely generated. These theorems extend the results of G. A. Margulis [Invent. Math. 76, 93-120 (1984; Zbl 0551.20028)] to positive characteristics.
Reviewer: A.D.Mednych

MSC:
22E40 Discrete subgroups of Lie groups
22E50 Representations of Lie and linear algebraic groups over local fields
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