The Haagerup property, Property (T) and the Baum-Connes conjecture for locally compact Kac-Moody groups

We indicate which symmetrizable locally compact affine or hyperbolic Kac-Moody groups satisfy Kazhdan’s Property (T), and those that satisfy its strong negation, the Haagerup property. This reveals a new class of hyperbolic Kac-Moody groups satisfying the Haagerup property, namely symmetrizable locally compact Kac-Moody groups of rank 2 or of rank 3 noncompact hyperbolic type. These groups thus satisfy the strongest form of the BaumConnes conjecture, namely the conjecture with coefficients in any C∗-algebra. For symmetrizable locally compact Kac-Moody groups G of rank 3 compact hyperbolic type or of affine or hyperbolic type and rank ≥ 4 we show that G has Property (T) and we deduce that the Baum-Connes assembly map on equivariant K-homology of G is both injective and surjective. Thus G satisfies the Baum-Connes conjecture without coefficients. We show that Property (T) and the Haagerup property for symmetrizable locally compact affine or hyperbolic Kac-Moody groups can be determined from the Dynkin diagram or equivalently from the generalized Cartan matrix. Our results give a dichotomy for hyperbolic Kac-Moody groups of noncompact type, with rank 3 Kac-Moody groups of noncompact hyperbolic type such as  1 satisfying the Haagerup property and hence the Baum-Connes conjecture with coefficients, and groups of rank 4 ≤ r ≤ 10, such as E10, satisfying Property (T) and the Baum-Connes conjecture without coefficients. For certain Kac-Moody lattices Γ satisfying the Haagerup property, we exhibit a proper action of Γ on a simplicial tree that is embedded in a space with a Lorentzian metric. The author was supported in part by NSF grant DMS-0701176 Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey,110 Frelinghuysen Rd Piscataway, NJ 08854-8019, USA E-mail: carbonel @ math.rutgers.edu