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Pansu, Pierre

Formulas of Matsushima, of Garland and property (T) for groups acting on symmetric spaces or buildings. (Formules de Matsushima, de Garland et propriété (T) pour des groupes agissant sur des espaces symétriques ou des immeubles.) (French) Zbl 0933.22009

Bull. Soc. Math. Fr. 126, No. 1, 107-139 (1998).
There is a whole bunch of so-called vanishing theorems, associated with the names of Matsushima and Garland. The Matsushima type theorems concern locally symmetric spaces, while Garland’s theorem relates to buildings. The author presents a very instructive survey of and geometric explanations for such results and gives new proofs for one variant of the Matsushima theorem and for Garland’s theorem. The main emphasis of the paper, however, is to show that these formulae yield Kazhdan’s property \((T)\). Thus let \(X\) be an irreducible symmetric space (distinct from the real or complex hyperbolic space) or a triangle building. Suppose that \(\Gamma\) is a discrete and cocompact group of isometries of \(X\), and let \(\rho\) be any unitary representation of \(\Gamma\). Then the first and second cohomology groups associated to \(\rho\) have the property that \(H^1(\Gamma,\rho)=0\) and \(H^2 (\Gamma, \rho)\) is separated. In particular, \(\Gamma\) has property \((T)\) since this is equivalent to that \(H^1(\Gamma,\rho)=0\) for all unitary representations \(\rho\) of \(\Gamma\).
Reviewer: E.Kaniuth (Paderborn)

Cited in 1 Review
Cited in 16 Documents

MSC:

22D10 Unitary representations of locally compact groups
22E40 Discrete subgroups of Lie groups
51E24 Buildings and the geometry of diagrams

Keywords:

Kazhdan’s property \((T)\); vanishing theorems; Matsushima type theorems; locally symmetric spaces; Garland’s theorem; buildings; unitary representation; cohomology groups
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References:

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