New Kazhdan groups

Introduction A locally compact, second countable group G is called Kazhdan if for any unitary representation of G, the first continuous cohomology group is trivial H 1 ct (G, ρ) = 0. There are several other equivalent definitions, the reader should consult [6], esp. 1.14 and 4.7. For some time now Kazhdan groups have attracted attention. One of the main challenges is to understand them geometrically. Recently Pansu [7], ˙ Zuk [10], and Ballmann– ´ Swi¸atkowski [1], went back to Garland's paper [5], improved it in several respects and produced among other things new examples of Kazhdan's groups. These examples, especially those in [1], are explicit and significantly different from classical ones. We also go back to Garland's paper, but instead of euclidean buildings, we study hyperbolic ones. An interesting class of hyperbolic buildings with cocompact groups of automorphisms were constructed by Tits [9]. He associates with a ring Λ and a generalised Cartan matrix M a Kac–Moody group. These groups provide BN pairs for buildings. A special case of particular interest to us is that of Λ a finite field and generalised Cartan matrix coming from hyperbolic reflection groups for which the fundamental domain is a simplex; there are 10 of them in dimension 2, two in dimension 3 and one in dimension 4. Buildings associated to these data are locally finite and their automorphism groups are locally compact topological groups. It turns out that they are Kazhdan' (and more). 1 Theorem 1. Let X q be an n-dimensional building of thickness (q + 1), associated to a cocompact hyperbolic group with the fundamental domain a simplex. Suppose G is a closed in the compact open topology, unimodular subgroup of the simplicial automorphism group which acts cocompactly on the building. Then for large q and 1 ≤ k ≤ n − 1 H k ct (G, ρ) = 0, that is the continous cohomology groups of G with coefficients in any unitary representation vanish. In particular G, considered as a topological group is a Kazhdan group. Several comments are in order: 1. The theorem holds for any hyperbolic building. However at present Tits' Kac–Moody buildings are the only examples where we can verify the assumptions. 2. For Tits' Kac–Moody building, simplicial automorphism groups which are uncountable, are bigger than Kac–Moody groups (given by countably many generators and relations), and Tits' Kac–Moody groups are not discrete as subgroups of automorphism …