Isometric group actions on Hilbert spaces: structure of orbits

Yves de Cornulier, Romain Tessera, Alain Valette

February 17, 2006

Abstract Our main result is that a finitely generated nilpotent group has no isometric action on an infinite dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

Mathematics Subject Classification: Primary 22D10; Secondary 43A35, 20F69. Key words and Phrases: Affine actions, Hilbert spaces, minimal actions, nilpotent groups.

1 Introduction

The study of isometric actions of groups on affine Hilbert spaces has, in recent years, found applications ranging from the K-theory of C^{∗}-algebras [HiKa], to rigidity theory [Sh2] and geometric group theory [Sh3, CTV]. This renewed interest motivates the following general problem: How can a given group act by isometries on an affine Hilbert space?

This paper is a sequel to [CTV], but can be read independently. In [CTV], given an an isometric action of a finitely generated group G on a Hilbert space α : G → Isom(H), we focused on the growth of the function g ↦ α(g)(0). Here the emphasis is on the structure of orbits.

We will mainly focus on actions of nilpotent groups. Let us begin by a simple example: every isometric action of Z on a Euclidean space is the direct sum of an action with a fixed point and an action by translations. This actually remains true for general locally compact nilpotent groups. The situation becomes more subtle when we study action on infinite-dimensional Hilbert spaces. However, something remains from the finite-dimensional case.

We say that a convex subset of a Hilbert space is locally bounded if its intersection with any finite dimensional subspace is bounded. The main result of the paper is the following theorem.

Theorem 1. (see Corollary 3.9 and Theorem 4.3) Let G be a locally compact, second countable, nilpotent group. Let G act isometrically on a Hilbert space H, with linear part π. Let O be an orbit under this action. Then there exist

• a subspace T of H (the “translation part”), contained in the invariant vectors of π, and

• a closed, locally bounded convex subset U of the orthogonal subspace T^{⊥},

such that O is contained in T × U.

We owe the following general question to A. Navas: which locally compact groups have an isometric action on a infinite-dimensional separable Hilbert space with dense orbits (i.e. minimal)?

Theorem 1 allows us to provide a negative answer in the case of finitely generated nilpotent groups.

Theorem 2. (see Corollary 4.6) A compactly generated, nilpotent-by-compact locally compact group does not admit any affine isometric action with dense orbits on an infinite-dimensional Hilbert space.

Actually, for compactly generated nilpotent groups, one can describe all affine isometric actions with dense orbits; see Corollary 4.5.

In the course of our proof, we introduce the following new definition: a unitary or orthogonal representation π of a group is strongly cohomological if it satisfies: for every nonzero subrepresentation ρ ≤ π, we have H^{1}(G, ρ) ≠ 0. It is easy to observe that the linear part of a affine isometric action with dense orbits is strongly cohomological. The non-trivial step in the proof of the main theorem is the following result.

Proposition 3. (see Corollary 3.9) Let π be an orthogonal or unitary representation of a second countable, nilpotent locally compact group G. Suppose that π is strongly cohomological. Then π is a trivial representation.

Another case for which we have a negative answer is the following.

Theorem 4. (see Theorem 4.7) Let G be a connected semisimple Lie group. Then G has no isometric action on a nonzero Hilbert space with dense orbits.

It is not clear how the main theorem can be generalized, in view of the following example.

Proposition 5. (see Proposition 2.1) There exists a finitely generated metabelian group admitting an affine isometric action with dense orbits on ℓ_{R}^{2}(Z).

Another construction provides

Proposition 6. (see Proposition 2.3) There exists a countable group admitting an affine isometric action with dense orbits on an infinite dimensional Hilbert space, in such a way that every finitely generated subgroup has a fixed point.

Acknowledgements. We thank A. Navas for useful discussions and encouragement.

2 Existence results

Here is a first positive result regarding Navas’ question.

Proposition 2.1. There exists an isometric action of a metabelian 3-generator group on a infinite-dimensional separable Hilbert space, all of whose orbits are dense.

Proof. Observe that Z[\sqrt{2}] acts by translations, with dense orbits, on R; so the free abelian group of countable rank Z[\sqrt{2}]^{(Z)} acts by translations, with dense orbits, on ℓ_{R}^{2}(Z). Observe now that the latter action extends to the wreath product Z[\sqrt{2}] ≀ Z = Z[\sqrt{2}]^{(Z)} ⋊ Z, where Z acts on ℓ_{R}^{2}(Z) by the shift. That wreath product is metabelian, with 3 generators.

Corollary 2.2. There exists an isometric action of a free group of finite rank on a Hilbert space, with dense orbits.

Recall that an isometric action α : G → Isom(H) almost has fixed points if for every ε > 0 and every compact subset K ⊂ G there exists v ∈ H such that sup_{g∈K} ∥v − α(g)v∥ ≤ ε.

In the example given by Proposition 2.1, the given isometric action clearly does not almost have fixed points, i.e. it defines a non-zero element in reduced 1-cohomology. The next result shows that this is not always the case.

Proposition 2.3. There exists a countable group Γ with an affine isometric action α on a Hilbert space, such that α has dense orbits, and every finitely generated subgroup of Γ has a fixed point. In particular, the action almost has fixed points.

Proof. We first construct an uncountable group G and an affine isometric action having dense orbits and almost having fixed points.

In H = ℓ_{R}^{2}(N), let A_{n} be the affine subspace defined by the equations

x_{0} = 1, x_{1} = 1, ..., x_{n} = 1,

and let G_{n} be the pointwise stabilizer of A_{n} in the isometry group of H. Let G be the union of the G_{n}’s. View G as a discrete group.

It is clear that G almost has fixed points in H, since any finite subset of G has a fixed point. Let us prove that G has dense orbits. Claim 1. For all x, y ∈ H, we have lim_{n→∞} |d(x, A_{n}) − d(y, A_{n})| = 0.

By density, it is enough to prove Claim 1 when x, y are finitely supported in ℓ_{R}^{2}(N). Take x = (x_{0}, x_{1}, ..., x_{k}, 0, 0, ...) and choose n > k. Then

d(x, A_{n})^{2} = ∑_{j=0}^{k}(x_{j} − 1)^{2} + ∑_{j=k+1}^{n} 1^{2} = n + 1 − 2 ∑_{j=0}^{k} x_{j} + ∑_{j=0}^{k} x_{j}^{2},

so that d(x, A_{n}) = \sqrt{n} + O(\frac{1}{\sqrt{n}}), which proves Claim 1.

Denote by p_{n} the projection on the closed convex set A_{n}, namely p_{n}(x_{0}, x_{1}, …) = (1, 1, …, x_{n+1}, x_{n+2}, …). Claim 2. For all x, y ∈ H, we have lim_{n→∞} ∥p_{n}(x) − p_{n}(y)∥ = 0.

This is a straightforward computation. Claim 3. G has dense orbits in H.

Observe that two points x, y ∈ H are in the same G_{n}-orbit if and only if d(x, A_{n}) = d(y, A_{n}) and p_{n}(x) = p_{n}(y). Fix x_{0}, z ∈ H. We want to show that lim_{n→∞} d(G_{n}x_{0}, z) = 0. So fix ε > 0. By the second claim, for some n_{0}, ∥p_{n}(x_{0}) − p_{n}(z)∥ ≤ ε/2 whenever n ≥ n_{0}. Set

W = {x ∈ H| p_{n}(x) = p_{n}(z)};

this is the orthogonal affine subspace of A_{n} passing through z. Then y_{0} = x_{0} + (p_{n}(z) − p_{n}(x_{0})) ∈ W. By the first claim, there exists n_{1} ≥ n_{0} such that |d(y_{0}, A_{n}) − d(z, A_{n})| ≤ ε/2 for every n ≥ n_{1}. Therefore there exists y ∈ W such that ∥y − z∥ ≤ ε/2 and d(y, A_{n}) = d(y_{0}, A_{n}) = d(x_{0}, A_{n}). By the previous observation, there exists g ∈ G_{n} such that y = gy_{0}. Then

d(gx_{0}, z) ≤ d(gx_{0}, gy_{0}) + d(gy_{0}, z) ≤ ε,

so that d(G_{n}x_{0}, z) ≤ ε for every n ≥ n_{1}, proving the last claim.

Using separability of H, it is now easy to construct a countable subgroup Γ of G also having dense orbits on H. Question 1. Does there exist an affine isometric action of a finitely generated group on a Hilbert space, having dense orbits and almost having fixed points?

3 Cohomology of unitary representations of nilpotent groups

Our non-existence results concerning nilpotent locally compact groups will be based on the following study of their unitary representations.

Definition 3.1. If G is a topological group and π a unitary representation, we say that π is strongly cohomological if every nonzero subrepresentation of π has nonzero first cohomology.

The following Lemma is Proposition 3.1 in Chapitre III of [Gu2].

Lemma 3.2. Let π be a unitary representation of G that does not contain the trivial representation. Let z be a central element of G. Suppose that 1 − π(z) has a bounded inverse (equivalently, 1 does not belong to the spectrum of π(z)). Then H^{1}(G, π) = 0.

Proof. If g ∈ G, expanding the equality b(gz) = b(zg), we obtain that (1 − π(z))b(g) is bounded by 2∥b(z)∥, so that b is bounded by 2∥(1 − π(z))^{−1}∥∥b(z)∥.

Lemma 3.3. Let G be a locally compact, second countable group, and π a strongly cohomological representation. Then π is trivial on the centre Z(G).

Proof. Fix z ∈ Z(G). As G is second countable, we may write π = ∫ ⊕ ; Ĝ ρ dμ(ρ), a disintegration of π as a direct integral of irreducible representations. Let χ : Ĝ → S^{1} : ρ ↦ ρ(z) be the continuous map given by the value of the central character of ρ on z. For ε > 0, set X_{ε} = {ρ ∈ Ĝ : |χ(ρ) − 1| > ε} and π_{ε} = ∫_{X_{ε}}^{⊕} ρ dμ(ρ), so that π_{ε} is a subrepresentation of π. Since |ρ(z) − 1|^{−1} < ε^{−1} for ρ ∈ X_{ε}, the operator

(π_{ε}(z) − 1)^{−1} = ∫_{X_{ε}}^{⊕}(ρ(z) − 1)^{−1} dμ(ρ)

is bounded. We are now in position to apply Lemma 3.2, to conclude that H^{1}(G, π_{ε}) = 0. By definition, this means that π_{ε} is the zero subrepresentation, meaning that the measure μ is supported in Ĝ−X_{ε}. As this holds for every ε > 0, we see that μ is supported in {ρ ∈ Ĝ : ρ(z) = 1}, to the effect that π(z) = 1.

Proposition 3.4. Let G be a topological group, and π a unitary representation of G. Suppose that \overline{H^{1}}(G, π) ≠ 0. Then π has a nonzero subrepresentation that is strongly cohomological.

Proof. Suppose the contrary. Then, by an standard application of Zorn’s Lemma, π decomposes as a direct sum π = ⨁_{i∈I} π_{i}, where H^{1}(G, π_{i}) = 0 for every i ∈ I, so that \overline{H^{1}}(G, π) = 0 by Proposition 2.6 in Chapitre III of [Gu2].

Remark 3.5. The converse is false, even for finitely generated groups: indeed, it is easy to check (see [Gu1]) that every nonzero representation of the free group F_{2} has non-vanishing H^{1}, so that every unitary representation of F_{2} is strongly cohomological. But it turns out that F_{2} has an irreducible representation π such that \overline{H^{1}}(F_{2}, π) = 0 (see Proposition 2.4 in [MaVa]).

Corollary 3.6. Let G be a locally compact, second countable group, and let π be a unitary representation of G without invariant vectors. Write π = π_{0} ⊕ π_{1}, where π_{1} consists of the Z(G)-invariant vectors. Then

(1) π_{0} does not contain any strongly cohomological subrepresentation (in particular, \overline{H^{1}}(G, π_{0}) = 0);

(2) every 1-cocycle of π_{1} vanishes on Z(G), so that H^{1}(G, π_{1}) ≃ H^{1}(G/Z(G), π_{1}).

Proof. (1) follows by combining Lemma 3.3 and Proposition 3.4. For (2), we use the idea of proof of Theorem 3.1 in [Sh2]: if b ∈ Z^{1}(G, π_{1}), then for every g ∈ G, z ∈ Z(G),

π_{1}(g)b(z) + b(g) = b(gz) = b(zg) = b(g) + b(z)

as π_{1}(z) = 1. So π_{1}(g)b(z) = b(z); this forces b(z) = 0 as π has no G-invariant vector. So b factors through G/Z(G).

Observe that Corollary 3.6 provides a new proof of Shalom’s Corollary 3.7 in [Sh2]: under the same assumptions, every cocycle in Z^{1}(G, π) is almost cohomologous to a cocycle factoring through G/Z(G) and taking values in a subrepresentation factoring through G/Z(G).

From Corollary 3.6 we immediately deduce

Corollary 3.7. Let G be a locally compact, second countable, nilpotent group, and let π be a representation of G without invariant vectors. Let (Z_{i}) be the ascending central series of G (Z_{0} = {1}, and Z_{i} is the centre modulo Z_{i−1}). Let σ_{i} denote the subrepresentation of G on the space of Z_{i}-invariant vectors, and finally let π_{i} be the orthogonal of σ_{i+1} in σ_{i}, so that π = ⨁ π_{i}.

Then H^{1}(G, π_{i}) ≃ H^{1}(G/Z_{i}, π_{i}) for all i, and π is not a strongly cohomological subrepresentation. In particular, \overline{H^{1}}(G, π) = 0. □

Note that the latter statement is a result of Guichardet [Gu1, Théorème 7], which can be stated as: G has Property H_{T} (i.e. every unitary representation with non-vanishing reduced cohomology contains the trivial representation).

Definition 3.8. We say that a locally compact group G has Property H_{CT} if every strongly cohomological unitary representation of G is trivial.

It is a straightforward verification that this is equivalent to: every strongly cohomological orthogonal representation of G is trivial. This will be useful in the next paragraph since we will deal with orthogonal rather than unitary representations.

As a corollary of Proposition 3.4, Property H_{CT} implies Property H_{T}. We have proved

Proposition 3.9. If G is a locally compact, second countable nilpotent group, then G has Property H_{CT}.

4 Non-existence results

Definition 4.1. 1) We say that subset Y of a metric space (X, d) is coarsely dense if there exists C ≥ 0 such that, for every x, y ∈ X,

d(x, G.y) ≤ C.

2) We say that a subset Y of a Hilbert space H is enveloping if its closed convex hull is all of H.

Observe that every dense subset of a metric space is coarsely dense. Besides, in a Hilbert space H, every coarsely dense subset Y is enveloping. Indeed, suppose that Y is contained in a closed, convex proper subset X of H. Consider v ∉ X and let y denote its projection on X (excluding the trivial case Y = ∅). Then, for every λ ≥ 0, we have d(y + λ(v − y), Y) ≥ d(y + λ(v − y), X) = λ, which is unbounded, so that Y is not coarsely dense.

Example 1. In ℓ_{R}^{2}(Z), let X denote the elements with integer coefficients. Then X is enveloping: indeed, its intersection with the subspace F_{n} = ℓ_{R}^{2}({−n, …, n}) is coarsely dense, hence enveloping in F_{n}, and the increasing union ⋃ F_{n} is dense in ℓ_{R}^{2}(Z). But X is not coarsely dense: indeed, for every n ≥ 0, the element \frac{1}{2}1_{\{1,...,4n\}} is at distance \sqrt{n} to X.

Note that X is the orbit of 0 for the natural action of the wreath product Z ≀ Z = Z^{(Z)} ⋊ Z on ℓ_{R}^{2}(Z), where Z^{(Z)} acts by translations and the factor Z acts by shifting (compare the example in the proof of Proposition 2.1).

Lemma 4.2. Let G be a topological group and π an orthogonal representation, admitting a 1-cocycle b with enveloping orbits. Then π is strongly cohomological.

Proof. If σ is a nonzero subrepresentation of π, let b_{σ} be the orthogonal projection of b on H_{σ}, so that b_{σ} ∈ Z^{1}(G, σ). Then b_{σ}(G) is enveloping in H_{σ}, in particular b_{σ} is unbounded. So b_{σ} defines a non-zero class in H^{1}(G, σ).

Theorem 4.3. Let G be a locally compact group with Property H_{CT}. Let G act isometrically on a Hilbert space H, with linear part π. Let O be an orbit under this action. Then there exist

• a subspace T of H, contained in H^{π(G)}, and

• a closed, locally bounded convex subset U of T^{⊥},

such that O is contained in T × U.

Proof. We immediately reduce to the case when π has no invariant vectors, so that we must prove that the closed convex hull U of O is locally bounded.

Observe that a convex subset of a Hilbert space is locally bounded if and only if it contains no affine half-line. Thus denote by D the set of affine half-lines contained in U, and suppose by contradiction that D ≠ ∅. Denote by D_{0} the corresponding set of linear half-lines (where the linear half-line corresponding to a half-line x + R_{+}v is simply R_{+}v). Then D_{0} is invariant under the linear action π of G. Let W be the closed subspace of H generated by all the half-lines in D_{0}, and denote by σ the corresponding subrepresentation. By assumption, σ is non-zero.

We claim that σ is strongly cohomological, contradicting that π has no invariant vectors along with the H_{CT} assumption. Let ρ be a non-zero subrepresentation of σ. Then by the definition of W, there exists an half-line of U which projects injectively into the subspace of ρ. Thus H^{1}(G, ρ) ≠ 0, proving the claim, and ending the proof.

Corollary 4.4. Let G be a locally compact group with Property H_{CT}. Let H be a Hilbert space on which G acts with with enveloping (respectively coarsely dense, resp. dense) image. Then the action is by translations, defined by a continuous morphism: u : G → (H, +) with enveloping (resp. coarsely dense, resp. dense) image.

Corollary 4.5. Let G be a locally compact, compactly generated group with Property H_{CT}, and let H be a (real) Hilbert space. Then • G has a isometric action on H with coarsely dense (respectively enveloping)

orbits if and only H has finite dimension k, and G has a quotient isomorphic to R^{n} × Z^{m}, with n + m ≥ k.

• G has a isometric action on H with dense orbits if and only H has finite dimension k, and G has a quotient isomorphic to R^{n} × Z^{m}, with max(n + m − 1, n) ≥ k.

Proof. Let α be an affine isometric action of G with enveloping orbits (this encompasses all possible assumptions). By Corollary 4.4, the action is by translations; let u be the morphism G → (H, +); its image generates H as a topological vector space. Let W denote the kernel of u.

Then A = G/W is a locally compact, compactly generated abelian group, which embeds continuously, in a Hilbert space. By standard structural results, A has a compact subgroup K such that A/K is a Lie group. Since K embeds in a Hilbert space, it is necessarily trivial, so that A is an abelian Lie group without compact subgroup. Accordingly, A is isomorphic to R^{n} × Z^{m} for some integers n, m; the embedding of A into H extends canonically to a linear mapping of R^{n+m} into H. In particular H is finite-dimensional, of dimension k ≤ n + m.

If the action has dense orbits, then either m = 0 and n ≥ k, or m ≥ 1 and m ≥ k − n + 1; this means that k ≤ max(n + m − 1, n). Conversely, if k ≤ n+m−1, then, since Z has a dense embedding in the torus R^{k}/Z^{k}, Z^{k+1} has a dense embedding in R^{k}, and this embedding can be extended to R^{n} × Z^{m}.

From Proposition 3.9 and Corollary 4.5, we deduce

Corollary 4.6. A compactly generated, nilpotent-by-compact group does not admit any isometric action with enveloping (e.g. dense) orbits on an infinitedimensional Hilbert space.

Proof. The only thing we have to care now is that the group G is not necessarily second countable. So let α be an isometric action with enveloping orbits on a Hilbert space H. By the Kakutani-Kodaira Theorem [Com, Theorem 3.7], there exists a compact normal subgroup N such that G/N is second countable. Since N is compact, the affine subspace α^{N} of N-fixed points is non-empty. Since it is G-invariant and the orbits are enveloping, necessarily α^{N} = H, so we are reduced to the case when G is second countable, allowing us to use Proposition 3.9 to conclude.

Proposition 2.1 on the one hand, and Corollary 4.6 on the other, isolate the first test-case for Navas’ question:

Question 2. Does there exist a polycyclic group admitting an affine isometric action with dense orbits on an infinite-dimensional Hilbert space?

Let us prove a related result for semisimple groups.

Theorem 4.7. Let G be a connected, semisimple Lie group. Then G cannot act on a Hilbert space H ≠ 0 with coarsely dense (e.g. dense) orbits.

Proof. Suppose by contradiction the existence of such an action α, and let π denote its linear part. Then π is strongly cohomological. By Lemma 3.3, π is trivial on the centre of G. Thus the centre acts by translations, generating a finitedimensional subspace V of H. The action induces a map p : G → O(V)⋉V. Since G is semisimple, the kernel of p contains the sum G_{nc} of all noncompact factors of G, and thus factors though the compact group G/G_{nc}. Thus H^{1}(G, V) = 0, and since π is strongly cohomological, this implies that V = 0.

It follows that α is trivial on the centre of G, so that we can suppose that G has trivial centre. Then G is a direct product of simple Lie groups with trivial centre. We can write G = H × K where K denotes the sum of all simple factors S of G such that α(S)(0) is bounded (in other words, H^{1}(S, π|_{S}) = 0). Then the restriction of α to H also has coarsely dense orbits. Moreover, every simple factor of H acts in an unbounded way, so that, by a result of Shalom [Sh1, Theorem 3.4]^{1}, the action of H is proper. That is, the map i : H → H given by i(h) = α(h)(0) is metrically proper and its image is coarsely dense. By metric properness, the subset X = i(H) ⊂ H satisfies: X is coarsely dense, and every ball in X (for the metric induced by H) is compact.

Suppose that H is infinite dimensional and let us deduce a contradiction. For some d > 0, we have d(x, X) ≤ d for every x ∈ H. If H is infinite dimensional, there exists, in a fixed ball of radius 7d, infinitely many pairwise disjoint balls B(x_{n}, 3d) of radius 3d. Taking a point in X ∩ B(x_{n}, 2d) for every n, we obtain a closed, infinite and bounded discrete subset of X, a contradiction.

Thus H is finite dimensional; since every simple factor of H is non-compact, it has no non-trivial finite dimensional orthogonal representation, so that the action is by translations, and hence is trivial, so that finally H = {0}.

Remark 4.8. 1) The same argument shows that a semisimple, linear algebraic group over any local field, cannot act with coarsely dense orbits on a Hilbert space.

2) The argument fails to work with enveloping orbits: indeed, in ℓ_{R}^{2}(N), let X denote the set sequences (x_{n}) such that x_{n} ∈ 2^{n}Z for every n ∈ N. Then X is coarsely dense in ℓ_{R}^{2}(N), but, for the metric induced by H, every ball in X is finite, hence compact. We do not know if a semisimple Lie group (e.g. SL_{2}(R)) can act isometrically on a non-zero Hilbert space with enveloping orbits.

1Shalom only states the result for a simple group, but the proof generalizes immediately.

See for instance [CLTV] for another proof, based on the Howe-Moore Property.

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Yves de Cornulier École Polytechnique Fédérale de Lausanne (EPFL) Institut de Géométrie, Algèbre et Topologie (IGAT) CH-1015 Lausanne, Switzerland

E-mail: decornul@clipper.ens.fr

Romain Tessera Équipe Analyse, Géométrie et Modélisation Université de Cergy-Pontoise, Site de Saint-Martin 2, rue Adolphe Chauvin F 95302 Cergy-Pontoise Cedex, France

E-mail: tessera@clipper.ens.fr

Alain Valette Institut de Mathématiques - Université de Neuchâtel Rue Emile Argand 11, CH-2007 Neuchâtel - Switzerland

E-mail: alain.valette@unine.ch