HAAGERUP PROPERTY FOR SUBGROUPS OF SL_{2} AND RESIDUALLY FREE GROUPS

YVES DE CORNULIER

Abstract. In this note, we prove that all subgroups of SL(2, R) have the Haagerup property if R is a commutative reduced ring. This is based on the case case when R is a field, recently established by Guentner, Higson, and Weinberger. As an application, residually free groups have the Haagerup property.

A locally compact, σ-compact group G is said to have the Haagerup property if it has a metrically proper isometric action on some Hilbert space. A recent panorama of the Haagerup property is proposed in [CCJJV]. Here, we only need the following facts: the Haagerup property is closed under taking closed subgroups, finite direct products, and the two less trivial facts ([CCJJV], chap. 6):

• (1) If Γ is a lattice in G, then G has the Haagerup property if (and only if) Γ does.

• (2) If G is discrete, then G has the Haagerup property if and only all its finitely generated

subgroups do. Motivated by (2), we say that a discrete (not necessarily countable) group has the Haagerup property if all its finitely generated subgroups do (equivalently, if all its countable subgroups do)^{1}.

The purpose of this short note is to point out a straightforward generalization of the following theorem:

Theorem 1 (Guentner-Higson-Weinberger [GHW], 2003). Let K be a field, and G be a subgroup of SL(2, K). Then G has the Haagerup property (as a discrete group).

Using theorem 1, we obtain the following generalization.

Theorem 2. Let R be a reduced (= without nilpotent elements) commutative ring, and G be a subgroup of SL(2, R). Then G has the Haagerup property (as a discrete subgroup).

Proof: 1) Suppose R is a finite product of fields. Then it is a immediate consequence of theorem 1 (since the Haagerup property is stable under taking finite direct products and (closed) subgroups).

2) General case. We can suppose that G is finitely generated, hence that R is finitely generated as a ring. So R is Noetherian, hence has a finite number of minimal prime ideals 𝔭_{i}. Since R is reduced, ⋂ 𝔭_{i} = {0}, so that R embeds in ∏ R/𝔭_{i}, hence in the finite product ∏ K_{i}, where K_{i} = Frac(R/𝔭_{i}). So case 1 applies. ■

Remarks 3.

• The assumption that R is reduced cannot be dropped. For instance, SL(2, Z[t]/t^{2}) does not have the Haagerup property. Indeed, if H is the kernel of the natural morphism SL(2, Z[t]/t^{2}) → SL(2, Z), then H is infinite, while the pair (SL(2, Z[t]/t^{2}), H) has Kazhdan’s relative property (T). This can be seen by embedding it as a lattice in the Lie group SL(2, R[t]/t^{2}), which is isomorphic to SL(2, R) ⋉ 𝔰𝔩(2, R), where the action of SL_{2}(R) on the vector space V = 𝔰𝔩(2, R) is the adjoint action. This is the three-dimensional real irreducible representation of SL_{2}(R), so that it is well-known that (SL_{2}(R) ⋉ V, V) has Property (T); see, for instance, Chapter 1 in [BHV].

Date: October 31, 2004.

2000 Mathematics Subject Classification. Primary 20E26; Secondary 22D10, 20G35.

1This coincides with the right definition of Haagerup property for general locally compact groups: existence of a unitary C^{0} representation weakly containing the trivial representation, or, equivalently, existence of a net of definite positive normalized C^{0} functions, converging to 1, uniformly on compact subsets.

• The commutativity assumption cannot be dropped, even in theorem 1. Indeed, let H be the skew-field of Hamilton quaternions. Then SL(2, H) has infinite subgroups with Kazhdan’s property (T): recall that SL(2, H) ≃ SO(5, 1), the latter contains SO(5) as a subgroup, and it is well-known that SO(5) has infinite subgroups with property (T) (for instance, obtained by projecting an irreducible lattice of SO(5) × SO(2, 3)).

Here is an application of theorem 2. Recall that a group G is called residually free is it satisfies one of the (clearly) equivalent conditions:

• (i) For all x ∈ G\{1}, there exist a (nonabelian) free group F and a morphism f : G → F such that f(x) ≠ 1.

• (ii) G embeds in a product of free groups.

• (iii) G embeds in a product of free groups of finite rank.

Theorem 4. Let G be a residually free group. Then G has the Haagerup property.

Proof: It suffices to show that any product ∏_{i∈I} F_{i} of free groups of finite rank has the Haagerup property. But such a product embeds in ∏_{i∈I} SL(2, Z) = SL(2, Z^{I}). So this follows from theorem 2. ■

Remarks 5. The Haagerup property is not closed under infinite products (with the discrete topology). If it were, all residually finite groups would have the Haagerup property! For instance, the discrete group ∏_{i} SL(n, Z/p^{i}Z) does not have the Haagerup property if n ≥ 3, since it contains the infinite Kazhdan group SL(n, Z) as a subgroup. On the other hand, we do not know if the class of torsion-free groups with the Haagerup property is closed under infinite products.

Remarks 6. V. Guirardel pointed out to us that, using some nontrivial properties of residually free groups, theorem 4 follows directly from theorem 1. Indeed, a residually free group can be embedded in SL(2, R), where R is a finite product of fields. The first ingredient is that a residually free group can be embedded in a finite product of fully residually free groups. The second ingredient is that a fully residually free group can be embedded in the ultraproduct ^{∗}F_{2}, which embeds in SL(2,^{∗}Q), and ^{∗}Q is a field. For details and and many other interesting properties of (fully) residually free groups, see [CG].

References

[B] Benjamin Baumslag. Residually Free Groups. Proceedings of the London Mathematical Society. Third Series, XVII 402-418, 1967.

[BHV] Bachir Bekka, Pierre de la Harpe, Alain Valette. Kazhdan’s Property (T). Forthcoming book, currently

         available at http://poncelet.sciences.univ-metz.fr/~bekka/, 2004.

[CCJJV] Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, Alain Valette. Groups with the Haagerup Property. Birkhäuser, Progress in Mathematics 197, 2001.

[CG] Christophe Champetier, Vincent Guirardel. Limit groups as limits of free groups: compactifying the set of free groups. Preprint, 2003, available on ArXiv (math.GR/0401042).

[GHW] Erik Guentner, Nigel Higson, Shmuel Weinberger. The Novikov Conjecture for Linear Groups. Preprint, 2003, to appear in Publications de l’IHES.

Yves de Cornulier EPF Lausanne, IGAT

E-mail: decornul@clipper.ens.fr
Homepage: http://www.eleves.ens.fr/home/decornul/