A Note on Quotients of Word Hyperbolic Groups with Property (t) or Property (fa)

Every discrete group with Kazhdan’s Property (T) (resp. Property (FA)) is a quotient of a torsion-free, word hyperbolic group with Property (T) (resp. (FA)). Recall ([HV], [Ser]) that a countable discrete group G has Property (T) (resp. (FA)) if every isometric action of G on an affine Hilbert space (resp. on a simplicial tree) has a fixed point (or, equivalently, has bounded orbits). As immediate consequences of the definitions, Property (T) and (FA) are inherited by quotients and by extensions. By a result of Alperin and Watatani [HV, Chap. 6], Property (T) implies Property (FA). The converse is well-known to be false: using [Ser, Corollaire 2, p.90], the affine Coxeter group 〈x, y, z |x = y = z = (xy) = (xz) = (yz) = 1〉 has Property (FA). On the other hand, it has a subgroup of finite index isomorphic to Z, hence without Property (T). Therefore it does not have Property (T), since Property (T) is known to be inherited by subgroups of finite index (see [BHV, Section 2.6] for a direct proof from the definition given here). However, Property (FA) and (T) share some features. For instance [Ser, Théorème 15 p. 81], a countable group G has Property (FA) if and only if it satisfies the three following conditions: (i) G is finitely generated, (ii) G does not map onto Z, (iii) G does not decompose as a nontrivial amalgam. Shalom [Sha, Theorem 6.2] has proved a similar characterization for Property (T): a countable group G has Property (T) if it satisfies Conditions (i), (ii), and (iii’), where (i) and (ii) as above, and with (iii’) defined as: Cor(G) = {1G}, where the cortex Cor(G) of G is defined as the set of (isomorphism classes of) irreducible unitary representation which cannot be Hausdorff separated from the trivial representation 1G for the Fell topology (see [HV, Chap. 1]). Shalom [Sha, Theorem 6.7] has proved the following remarkable result for Property (T). Theorem 1 (Shalom, 2000). For every group G with Property (T), there exists a finitely presented group G0 with Property (T) which maps onto G. In other words, this means that, given a finite generating subset for G, only finitely many relations suffice to imply Property (T). This can be interpreted in the topology of marked groups [Cha] as: Property (T) is an open property. Relying on ideas of V. Lafforgue, we prove that a similar result holds for Property (FA). Theorem 2. For every group G with Property (FA), there exists a finitely presented group G0 with Property (FA) which maps onto G. We refer to [GH] for the notion, due to Gromov, of word hyperbolicity. We only recall that a word hyperbolic group is a finitely generated group whose Cayley graph satisfies a certain condition meaning that, at large scale, it is negatively curved. We only mention here that word hyperbolic groups are necessarily finitely presented, and that word hyperbolicity is a fundamental notion in combinatorial group theory as in geometric topology. It was asked [Wo, Question 16] whether every group with Property (T) is quotient of a group with Property (T) with finiteness conditions stronger than finite presentation. We give an answer here by showing that we can impose word hyperbolicity. Theorem 3. For every group G with Property (T) (resp. (FA)), there exists a torsion-free word hyperbolic group G0 with Property (T) (resp. (FA)) which maps onto G. Date: April 10, 2008. 2000 Mathematics Subject Classification. Primary 20F65; Secondary 20E08, 20F67.