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Rigidity of commensurators and irreducible lattices. (English) Zbl 0978.22010
Let $$G$$ be a compactly generated locally compact group of the form $$G_1\times G_2 \times \cdots \times G_n$$ and let $$\Gamma$$ be a discrete cocompact subgroup of $$G$$ (namely a uniform lattice in $$G$$) whose projection on each $$G_i$$ is dense in the latter. A new result is proved for unitary representations of $$\Gamma$$ relating to superrigidity, viz. extendibility of the representations to $$G$$, (too technical to be stated here in detail), and applied to various questions. One of the applications asserts that if $$\Delta$$ is a normal subgroup of $$\Gamma$$ then $$\Gamma/\Delta$$ has Kazhdan’s property (T) if and only if there are no nontrivial homomorphisms of $$G$$ into $${\mathbb C}$$ vanishing on $$N$$ and the quotient $$G_i/\overline{p_i(N)}$$ has property (T) for each $$i$$, where $$p_i$$ denotes the $$i$$th projection. Other applications concern superrigidity of $$\Gamma$$, and also of dense subgroups containing $$\Gamma$$ and contained in the commensurator of $$\Gamma$$, in the sense of extendibility of homomorphisms into certain groups to the whole of $$G$$. The latter in turn yields results on arithmeticity of the lattices and their commensurators. For $$G_i$$ arising from the usual algebraic group setup similar results are also proved for nonuniform lattices. The author notes that underlying the analysis in the paper is a basic observation showing that commuting isometric actions on a Hilbert space must exhibit a certain kind of “degeneracy”.
Reviewer: S.G.Dani (Mumbai)

##### MSC:
 2.2e+41 Discrete subgroups of Lie groups 2.2e+11 General properties and structure of complex Lie groups
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