Journal of Lie Theory Volume 13 (2003) 383–385 C ◯2003 Heldermann Verlag

New a-T-menable HNN-extensions

Światosław R. Gal and Tadeusz Januszkiewicz^{∗}

Communicated by A. Valette

Abstract. The Baumslag-Solitar groups are a-T-menable. This is proved by embedding them into topological groups and studying representation theoretic properties of the latter. The paper is motivated by a question of A. Valette. 2000 Mathematics Subject Classification: 20F65

We are mainly interested in Baumslag-Solitar groups which are discrete. However, topological groups (Lie groups and the automorphism group of a tree) arise in the proof.

Definition 1 (M. Gromov). A locally compact, second countable group G is a-T-menable iff there exists a metrically proper affine isometric action of G on some Hilbert space.

a-T-menability is often referred to as the Haagerup approximation property. It is equivalent to the existence in C_{0}(G) of an approximate unit of positive definite functions on G. For ample discussion one may consult [2].

Let G ⊂ 𝔑 be a closed subgroup of a locally compact compactly generated topological group 𝔑. Let i_{k}: H → G (k = 1, 2) be two inclusions onto finite index open subgroups, which are conjugated by an automorphism ϕ of 𝔑.

The main case of interest is when G is discrete and 𝔑 is a Lie group.

Definition 2. A 𝔑-BS group is the group derived from such (G, H, i_{1}, i_{2}) by the (topological) HNN construction. In other words, if G is given by the presentation ⟨S|R⟩, Γ has the presentation ⟨S, t|R, ti_{1}(g)t^{−1} = i_{2}(g) ∀g ∈ H⟩.

Through this paper let Γ be a 𝔑-BS group as above.

Since we are working with topological presentations, recall that the topology in the HNN extension Γ is given by the basis B = {γUγ′: U open in G, γ, γ′ ∈ Γ}. It is clear, that Γ is a topological group with respect to B.

To see that B is in fact a basis consider U open in G. We can decompose U with respect to i_{1}(H) as follows U = ⋃ i_{1}(U_{n})g_{n}, where g_{n} are representatives of cosets of i_{1}(H). Then tU = ⋃ i_{2}(U_{n})tg_{n}. Therefore any set of B can be

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag written uniquely in form ⋃ U_{γ}γ where U_{γ} runs over open subsets of G and γ runs over the chosen set of representatives of G\Γ. Thus B is closed under the intersections and in particular B is a basis. The same argument shows that the topology on G induced from Γ concides with the original one (i.e. G ⊂ Γ is an open embedding). Examples:

∗ Both authors were partially supported by a KBN grant 2 P03A 035 20.

1. (𝔑 = ℝ) Baumslag-Solitar group with parameters p and q is given by a following presentation BS_{q}^{p} = ⟨x, t|x^{p} = tx^{q}t^{−1}⟩ = HNN(ℤ, ℤ, p·, q·).

2. (𝔑 = ℝ^{n}) Torsion free, finitely generated, abelian-by-cyclic groups are exactly ascending (i_{1} = id) HNN extensions of ℤ^{n} with i_{2} given by a n by n matrix with nonzero determinant [1].

3. 𝔑 a homogeneous nilpotent group (i.e. one admitting a dilating automor-

phism ϕ) with a discrete subgroup G ⊂ 𝔑 such that ϕ(G) ⊂ G.

An obvious adaptation of the Bass-Serre theory [6] to the topological context shows that for a topological HNN extension Γ of a group G there is a tree T with an edge-transitive Γ-action such that the vertex stabilizers are conjugated to G and edge stabilizers are conjugated to H.

Since i_{k}(H) are of finite index in G, T is locally finite. So, the simplicial automorphism group Aut(T) carries the natural (compact-open) topology such that the group is locally compact. More precisely, the basis of neighborhoods of the identity is the family U_{K}: = {g|gv = v ∀v ∈ K} where K runs over the family of the compact subsets K ⊂ T. Denote j_{T}: Γ → Aut(T) the homomorphism given by the action.

Define 𝔑̃ = ℤffn𝔑, to be the semidirect product given by the ℤ-action on 𝔑 via ϕ. There is an obvious homomorphism j_{𝔑}: Γ → 𝔑̃, which is the identity on G and sends t to the generator of ℤ.

Theorem 1. Let Γ be a 𝔑-BS group, and let j_{T} and j_{𝔑} be the homomorphisms defined above. Then the homomorphism j = (j_{T} , j_{𝔑}): Γ →Aut(T) × 𝔑̃ is an embedding onto a closed subgroup which is a topological isomorphism onto its image. Proof. Observe that since j_{𝔑} restricted to G is an embedding onto a closed subgroup (this follows from the fact that G in closed subgroup in 𝔑) the same is true for j restricted to G. Let v be the vertex stabilized by G. Then j_{T} (G) = Stab_{v} ∩ j_{T} (Γ).

Let g ∈ Γ such that j(g) = 1. Since j_{T} (g) = 1, gv = v i.e. g ∈ G. Since j_{𝔑} restricted to G is an embedding; this implies g = 1. Thus injectivity follows.

Let γ ∉ j(Γ). Since Γ acts transitively on T, we can multiply γ by some element of j(Γ) and assume that γv = v. Then (Stab_{v} − G) × 𝔑 is an open neighbourhood of γ omitting j(Γ). So j(Γ) is closed.

The topologies on G and j(G) coincide. Since G and j(G) are open in Γ and j(Γ) respectively, the same is true for Γ and j(Γ).

Corollary 1. If 𝔑 is a-T-menable then so are 𝔑-BS groups. Proof. Since T is locally finite, the group Aut(T) is a-T-menable by a result of Haagerup [4].

A cyclic extension of a-T-menable group is a-T-menable by [5]. Therefore 𝔑̃ is a-T-menable.

It is clear that a-T-menability is closed under direct products and under taking closed subgroups. Thus the corollary follows from Theorem 1.

Remark 1. According to Chapter 4 of [2] every a-T-menable connected Lie group 𝔑 is locally isomorphic to a direct product of an amenable group and copies of SO(1, n) and SU(1, n). A Lie group is amenable if it is compact-by-solvable. Thus the examples 1.-3. are a-T-menable.

Remark 2. Another proof of Corollary 1, relying on the study of a-T-menable of amalgams, is given in a forthcoming paper by the first author [3].

References

[1] Bieri, R., and K. Strebel, Almost finitely presented soluble groups, Comm. Math. Helv. 53 (1978), 258–278.

[2] Cherix, P. A., M. Cowling, P. Jolissant, P. Julg, and A. Valette, “Groups with the Haagerup property (Gromov’s a-T-menability),” Birkhäuser Verlag, 2001.

[3]  Gal, S. R., a-T-menability of groups acting on trees, Preprint.

[4] Haagerup, U., An example of a non-nuclear C^{∗} -algebra which has the metric approximation property, Invent. Math. 50 (1979), 279–293.

[5] Jolissaint, P., Borel cocycles, approximation properties and relative property T, Ergodic Th. Dyn. Syst. 20 (2000), 483–499.

[6]  Serre, J. P., “Trees,” Springer-Verlag, 1980.

Światosław R. Gal Instytut Matematyczny Uniwersytet Wrocławski pl. Grunwaldzki 2/4 50-384 Wrocław sgal@math.uni.wroc.pl Tadeusz Januszkiewicz Instytut Matematyczny PAN and Instytut Matematyczny Uniwersytet Wrocławski pl. Grunwaldzki 2/4 50-384 Wrocław tjan@math.uni.wroc.pl

Received January 6, 2002 and in final form October 25, 2002