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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)

22E40 Discrete subgroups of Lie groups
22E10 General properties and structure of complex Lie groups
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