KAZHDAN PROPERTY FOR SPACES OF CONTINUOUS FUNCTIONS

YVES DE CORNULIER

Abstract. We combine results of Vaserstein and Shalom to prove Kazhdan’s Property (T) for various topological groups of the form SL(n, C(X, R)) or SL(n, C(X, C)), for n ≥ 3 and X a topological subspace of a Euclidean space.

All the rings here are unitary and commutative. If R is a ring, let E(n, R) denote the subgroup of SL(n, R) generated by elementary matrices. If E(n, R) is normal in SL(n, R), the quotient is denoted by SK_{1,n}(R).

If R is a topological ring, then E(n, R) and SL(n, R) are topological groups for the topology induced by the inclusion in R^{n^{2}}. We say that a topological ring is topologically finitely generated it it has a finitely generated dense subring.

For any topological spaces X, Y, we denote by C(X, Y) the set of all continuous functions X → Y. If K denotes R or C, C(X, K) is a topological ring for the compact-open topology, which coincides with the topology of uniform convergence on compact subsets.

It is known [Vas86] that, if n ≥ 3,

E(n, C(X, K)) = {u : X → SL(n, K) homotopically trivial}

(this is immediate if X is compact, for all n ≥ 2).

If G is a topological group, a continuous unitary representation of G on a Hilbert space H almost has invariant vectors if, for every compact (not necessarily Hausdorff) subset K ⊂ G and every ε > 0, there exists ξ ∈ H, of norm one, such that sup_{g∈K} ∥g · ξ − ξ∥ ≤ ε. The topological group G has Kazhdan’s Property (T) or Property (T) if every continuous unitary representation of G almost having invariant vectors actually has nonzero invariant vectors; see [BHV05] for more about Property (T).

Theorem 1. Let n ≥ 3, and X ⊂ R^{d} be a topological subspace of a Euclidean space. Let K = R or C. Endow C(X, K) with the topology of uniform convergence on compact subsets. Then E(n, C(X, K)) has Kazhdan’s Property (T).

Corollary 2. Let n ≥ 3, and X ⊂ R^{d} be a compact subset. Endow X with the topology of uniform convergence. Then SL(n, C(X, K)) has Kazhdan’s Property (T) if and only if the discrete group SK_{1,n}(C(X, K)) = SL(n, C(X, K))/E(n, C(X, K)) does.

Date: 19th October 2005.

2000 Mathematics Subject Classification. Primary 18F25; Secondary 22D10.

Proof: Since X is compact^{1}, E(n, C(X, K)), is open (hence closed) in SL(n, C(X, K)). The corollary follows from the trivial fact that Property (T) is stable under quotients and extensions. ■

Example 3. Fix k ≥ 1 and n ≥ 3. Then SK_{1,n}(C(S^{k}, K)) = π_{k}(SL(n, K)), which is an abelian group. It follows that SL(n, C(S^{k}, K)) has Kazhdan’s Property (T) if and only if π_{k}(SL(n, K)) is finite; it is known (see [MiTo91]) that it is infinite if and only if:

• K = C, k is odd, and 3 ≤ k ≤ 2n − 1, or

• K = R, (k ≡ −1 (mod 4) and 3 ≤ k ≤ 2n − 1) or (n is even and k = n − 1) In particular, π_{k}(SL(n, K)) is finite for k = 1, k even, or k ≥ 2n.

Example 4. Let W denote a Cantor set. It is straightforward to show that, for every connected manifold M, all maps W → M are homotopic. Thus, SK_{1,n}(C(W, K)) is trivial, and accordingly, for all n ≥ 3, SL(n, C(W, K)) has Kazhdan’s Property (T).

Theorem 1 rests on two results: a K-theoretic result of Vaserstein (Theorem 8), and the work of Shalom on Kazhdan’s Property (T) (Theorem 7).

In [Sha99], Shalom introduces new methods to establish Kazhdan’s Property (T) for the special linear groups over certain rings. This leads, for instance, to the first proof that Γ = SL(n, Z), n ≥ 3, has Property (T), that does not use the embedding of Γ into SL(n, R) as a lattice.

Before stating his main result, let us begin with a definition.

Definition 5. If G is a group and S ⊂ G is a subset, we say that G is boundedly generated by S if there exist m < ∞ such that every g ∈ G is a product of at most m elements in S.

Theorem 6 (Shalom, [Sha99]). Let n ≥ 3, and let R be a topologically finitely generated ring. Suppose that SL(n, R) is boundedly generated by elementary matrices. Then SL(n, R) has Kazhdan’s Property (T).

As an application, Shalom proves bounded elementary generation for the loop group SL(n, C(S^{1}, C)) (n ≥ 3), and deduces that it has Property (T). He asks if the same holds for R instead of C, noting that SL(n, C(S^{1}, R)) is not generated by elementary matrices (since π_{1}(SL(n, R)) ≠ 1). This is answered positively by Theorem 1; see Example 3 above.

Actually, without modification, the proof of Theorem 6 ([Sha99], see also [BHV05]) gives a stronger statement.

Theorem 7. Let n ≥ 3, let R be a topologically finitely generated commutative ring, and suppose that E(n, R) is boundedly generated by elementary matrices. Then E(n, R) has Kazhdan’s Property (T). ■

Theorem 6 is the particular case of Theorem 7 when E(n, R) = SL(n, R).

Theorem 7 is a strong motivation for studying bounded elementary generation for the group E(n, R). For instance, this is an open question for R = F_{p}[X, Y] or R = Z[X], for all n ≥ 3. We now focus on the case when R = C(X, K), where K = R or C.

1If X is not compact, we do not know whether E(n, C(X, K)) is closed in SL(n, C(X, K)) for the topology of uniform convergence on compact subsets. However, we can restate the corollary as follows: SL(n, C(X, K)) has Kazhdan’s Property (T) if and only if the group \overline{SK}_{1,n}(C(X, K)) = SL(n, C(X, K))/\overline{E(n, C(X, K)}) does.

The notion of dimension of a topological space involved here is defined in [Vas71], and it will be sufficient for our purposes to know that dim(X) is finite for every topological subspace of a Euclidean space.

Theorem 8 (Vaserstein, [Vas88]). Let X be a finite dimensional topological space, and let K = R or C, and fix n ≥ 3. Then E(n, C(X, K)) is boundedly generated by elementary matrices.

Now we show how Theorems 7 and 8 imply Theorem 1. Since X ⊂ R^{d} for some d, X is finite dimensional, so that Theorem 8 applies: E(n, C(X, K)) is boundedly generated by elementary matrices.

It remains to show that Theorem 7 applies, that is, C(X, K) is topologically finitely generated for the compact-open topology (that is, the topology of uniform convergence on compact subsets).

Let p_{1}, …, p_{d} be the projections of X on the d coordinates of R^{d}, and let A be the (unital) K-subalgebra of C(X, K) generated by p_{1}, …, p_{d}. By the Stone-Weierstrass Theorem, A is dense in C(X, K) for the topology of uniform convergence on compact subsets. Then the finite family \{(p_{j}), \sqrt{2}\} (resp. \{(p_{j}), \sqrt{2}, i\}) generates a dense subring in C(X, K) if K = R (resp. if K = C). ■

Remark 9.

(1) The hypothesis in Theorem 1 that X is homeomorphic to a subset of an Euclidean space is close to being necessary in order to apply Theorem 7. Indeed, suppose that C(X, K) is endowed with a topology such that the evaluation functions f ↦ f(x) are continuous. Besides, suppose that C(X, K) is topologically finitely generated as a ring by p_{1}, … p_{d}. Then there exists a continuous injection of X in some Euclidean space, given by x ↦ (p_{1}(x), …, p_{d}(x)).

(2) If X is metrizable and non-compact, and C(X, K) is endowed with the uniform convergence topology, then an easy growth argument shows that C(X, K) is not topologically finitely generated.

It would be interesting to generalize Theorem 8 to general semisimple Lie groups without compact factors, and Theorem 7 to semisimple groups without compact factors and with Property (T), or at least to higher rank ones. Theorem 6 is extended to symplectic groups in [Neu03]. On the other hand, if G is a connected compact simple Lie group, C(S^{1}, G) does not have Kazhdan’s Property (T) [BHV05, Exercise 4.4.5].

Remerciements. J’ai une pensée émue pour Michel Matthey qui nous a récemment quittés, et envers qui je suis très reconnaissant pour m’avoir communiqué ces articles de Vaserstein, et pour avoir gentiment répondu à toutes mes questions. Je remercie également Bachir Bekka, Pierre de la Harpe et Alain Valette pour leur relecture et les corrections utiles qui s’en sont suivies.

References

[BHV05] Bachir Bekka, Pierre de la Harpe, Alain Valette. “Kazhdan’s Property (T)”, forth-

        coming book, available at http://name.math.univ-rennes1.fr/bachir.bekka/, 2005.

[MiTo91] Mamoru Mimura, Hiroshi Toda. “Topology of Lie groups, I and II”. Providence, R.I., American Mathematical Society, 1991.

[Neu03] Markus Neuhauser. Kazhdan’s property (T) for the symplectic group over a ring. Bull. Belg. Math. Soc. Simon Stevin 10(4), 537-550, 2003.

[Sha99] Yehuda Shalom. Bounded generation and Kazhdan’s property (T). Publ. Math. Inst. Hautes Études Sci. 90, 145-168, 1999.

[Vas71] Leonid N. Vaserstein. Stable rank of rings and dimensionality of topological spaces. Funct. Anal. Appl. 5, 102-110, 1971.

[Vas86] Leonid N. Vaserstein. On K_{1}-Theory of Topological spaces; in “Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Part II”. AMS Providence, Contemporary Mathematics 55, 729-740, 1986.

[Vas88] Leonid N. Vaserstein. Reduction of a matrix depending on parameters to a diagonal form by addition operators. Proc. Amer. Math. Soc. 103, No 3, 741-746, 1988.

Yves de Cornulier ENS Paris; EPF Lausanne, IGAT École Polytechnique Fédérale de Lausanne (EPFL) Institut de Géométrie, Algèbre et Topologie (IGAT)

E-mail: decornul@clipper.ens.fr