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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:
 2.2e+41 Discrete subgroups of Lie groups 2.2e+51 Representations of Lie and linear algebraic groups over local fields
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