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On \(L^2\)-cohomology and property (T) for automorphism groups of polyhedral cell complexes. (English) Zbl 0897.22007
Let \(\Gamma\) be a locally compact group. A unitary representation \(\rho\) of \(\Gamma\) on a Hilbert space \(H\) is said to admit an almost invariant vector if for any compact subset \(K\) in \(\Gamma\) and any \(\varepsilon >0\) there exists a unit vector \(v\in H\) such that \(\|\rho (g)v- v\|< \varepsilon\) whenever \(g\in\Gamma\). The group is said to admit Kazhdan’s property \((T)\) if for any unitary representation the existence of an almost invariant vector implies the existence of an invariant unit vector. This class of groups was studied recently by P. de la Harpe and A. Valette [La propriété de Kazhdan pour les groupes localement compacts, Astérisque 175 (Paris 1989; Zbl 0759.22001)]. In case \(\Gamma\) admits a countable basis for its topology, property \((T)\) holds if and only if any continuous symmetric function of negative type on \(\Gamma\) is bounded, and if and only if the first cohomology group \(H^1 (H,\rho)\) is trivial for any unitary representation \(\rho\). The authors update and extend Garland’s work on cohomology groups [H. Garland, Ann. of Math., II. Ser. 97, 375-423 (1973; Zbl 0262.22010)]. They determine a large collection of groups with property \((T)\) and obtain also groups lacking this property. They bring in the cohomology \(L^2 H(X,\rho)\), i.e., the complex of \(\bmod \Gamma\) square integrable cochains on a simplex \(X\) which are twisted by \(\rho\).

22D10 Unitary representations of locally compact groups
22D05 General properties and structure of locally compact groups
05C99 Graph theory
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