Hilbert space compression under direct limits and certain group extensions

We find bounds on the Hilbert space compression of the limit of a directed metric system of groups. We also give estimates on the Hilbert space compression of group extensions of a group $H$ by a a word-hyperbolic group or a group of polynomial growth.


Introduction
In [5], Gromov introduced the notion of uniform embeddability of a finitely generated group into a Hilbert space and suggested that such a group would satisfy the Novikov Conjecture [6]. Six years later, Yu came up with a formal proof of this claim [13]. Moreover, together with Skandalis and Tu, he proved that such uniformly embeddable groups also satisfy the coarse Baum-Connes Conjecture [10]. Definition 1.1. A metric space (X, d) is uniformly embeddable in a Hilbert space, if there exist a Hilbert space H, non-decreasing functions ρ − , ρ + : R + → R + such that lim t→∞ ρ − (t) = +∞, and a map f : X → H, such that The map f is called a uniform embedding of X in H. It is called large-scale Lipschitz whenever ρ + can be taken of the form ρ + : t → Ct + D for some C > 0, D ≥ 0. It is Lipschitz if we can take D = 0.
Two length functions l 1 and l 2 on a group (see Definition 2.1) are coarsely equivalent if for every R > 0 there exists S > 0 such that the l 1 -ball B(1, R) with radius R and center 1 is contained in the l 2 -ball B(1, S); and conversely. Clearly in this case, (X, l 1 ) is uniformly embeddable if and only if (X, l 2 ) is uniformly embeddable. Lemma 2.1 in [12] shows that every discrete countable group X admits a unique proper length function up to coarse equivalence, enabling us to define the concept of discrete countable uniformly embeddable group. This class of groups and its permanence properties have been very well studied, for example by Guentner and Dadarlat in [3]. For our purposes and theirs, the following reformulation of their Proposition 2.1, which holds for any metric space, is vital. Proposition 1.2. Let (X, d) be a metric space. Then X is uniformly embeddable in a Hilbert space if and only if for every n > 0 there exist S n > 0 and a Hilbert space valued map ξ n : X → H, x → ξ x n such that ξ x n = 1 for all x ∈ X and such that The speed at which (S n ) n∈N0 tends to infinity is an indication on how uniformly embeddable a metric space really is. For example, if n → S n is bounded by a polynomial map in n, it would make sense to say that the corresponding space is more uniformly embeddable than a space for which n → S n is only bounded by an exponential map in n. Another, more standard way of describing how uniformly embeddable a metric space X really is, is by looking at the supremum of δ ≥ 0 such that there is a large-scale Lipschitz uniform embedding of X and numbers C ′ , D ′ > 0 such that ρ − in Definition 1.1 can be taken of the from r → 1 C ′ r δ − D ′ . This supremum is called the Hilbert space compression of X [7]. For X a group, it must be noted that Hilbert space compression is a quasi-isometric invariant, but no longer a coarse invariant. Therefore, it is important that we always specify the chosen length function. Looking closely at the proof of Proposition 1.2, one finds a connection between the Hilbert space compression of a metric space and the growth of the sequence (S n ) n∈N0 . In this note, we try roughly to exploit this connection and then use techniques from [3] to get concrete information about the behaviour of the Hilbert space compression of groups under taking direct limits and under taking certain group extensions. Regarding group extensions, we prove the following results in Section 3 (see Theorems 3.3 and 3.5). Theorem 1.3. Assume that Γ is a group, equipped with some length function l = l Γ , that fits in a short exact sequence Define a length function l G on G by setting l G (π(x)) = inf{l(y) | π(y) = π(x)}. If G with the induced metric from Γ has polynomial growth and if H with the induced metric from Γ has compression δ, then the compression of Γ is at least δ/4. Theorem 1.4. Assume that Γ is a finitely generated group, equipped with the word length function l = l Γ relative to some finite symmetric generating subset S and that it fits in a short exact sequence Equip G with the word length function l G relative to π(S). If G is a finitely generated hyperbolic group in the sense of Gromov [4] and if H, with the induced metric from Γ, has Hilbert space compression δ, then the Hilbert space compression of Γ is at least δ/5.
We assume that every metric space in this article is a group and we assume that the metric is induced by a length function.
2. Hilbert space compression for the limit of a directed system of groups Throughout this article, every metric space will be a group whose metric is induced by a length function. Let us start by recalling the definition of a length function on a group. Definition 2.1. A length function l on a group G is a function l : ∀x, y ∈ G, l(xy) ≤ l(x) + l(y). We say that l is proper, whenever Every length function on G induces a left-invariant metric on G by d(x, y) = l(x −1 y) ∀x, y ∈ G.
. be a directed system of groups such that the maps G i → G i+1 are isometric injections. Denote G the direct limit of this system. By definition, G can be seen as the disjoint union of all the G i divided by some equivalence relation. Define the induced length function l on G by l(x) := lim i l Gi (x). We proceed under the assumption that l is a proper length function on G. In this section, we ask ourselves the question how the Hilbert space compression of G, denoted by α(G), is related to the Hilbert space compressions of the G i .
To begin, notice that every G i can be seen as a metric subspace of G and so α(G) ≤ inf i∈N α(G i ). Clearly, this bound is sharp, since as a family of subgroups we can take G i = G (∀i ∈ N). It proves more challenging to find a good lower bound for α(G). First, note that the same bound as above, i.e. inf i∈I α(G i ), is not always a lower bound. As an example, equip the group with the induced metric from Z ≀ Z. This group is the direct limit of the family of subgroups Since this metric is quasi-isometric to the standard word length metric on Z 2n+1 , we obtain Z (Z) as a limit of groups with compression 1. However, it follows from the proof of Theorem 3.9 in [1] that Z (Z) has compression less than 3 4 . Notice moreover that Z and Z (Z) have different compressions although they are both limits of groups of compression 1. It will thus be necessary to include more information on how the groups G i are embedded in their respective Hilbert spaces in order to say something useful about the Hilbert space compression of their limit.
We propose the following Theorem 2.2. Assume that G is the direct limit of a directed metric system (G i ) i∈N of groups and that the induced length function l is proper. .
Clearly, then G is a directed system of metric spaces as above. Recalling the fact that finite groups have Hilbert space compression equal to 1, we can apply Theorem 2.2, obtaining α(G) = 1.
Example 2.4. Let G ≀ H be finitely generated and equip it with the word length metric relative to a finite symmetric generating subset. Theorem 2.2 can be used as an easy way to estimate the compressions of spaces G (H) := {f : H → G | f has finite support }, equipped with the induced length function from G ≀ H. If G is a discrete group with compression α and H has polynomial growth of order d, then we obtain the lower bound 2α d+4 . It must be mentioned that the so obtained lower bound is weaker than the lower bound obtained in [8]. Proposition 1.2 from the Introduction plays a very important role in our proofs. It is implied by the following Proposition, which is Proposition 2.1 of [3]. We will give a (slightly modified version of) Guentner and Dadarlat's proof here because the details will be of vital importance further on.
Proposition 2.5. Let X be a metric space. Then X is uniformly embeddable in a Hilbert space if and only if for every R > 0 and ǫ > 0 there exists a Hilbert space valued map ξ : Proof. Assume that X is uniformly embeddable and let F : X → H be a uniform embedding of X in a real Hilbert space H. Let ρ − and ρ + be functions such that and define Exp : H → Exp(H) by Note that Exp(ζ), Exp(ζ ′ ) = e ζ,ζ ′ , for all ζ, ζ ′ ∈ H. For t > 0 define , it is easy to verify conditions 1 and 2 above.
Conversely, choose p > 0 and assume that X satisfies the conditions in the statement. There exist a sequence of maps η n : X → H n and a sequence of numbers S 0 = 0 < S 1 < S 2 < . . ., increasing to infinity, such that for every n ≥ 1 and every x, It is not hard to verify that F is well defined and Sn) , and the χ [Sn−1,Sn) are the characteristic functions of the sets [S n−1 , S n ).
Indeed, let x, x ′ ∈ X. If n is such that The following corollary gives a connection between the compression of X and the growth of the (S n ) n∈N0 . Corollary 2.6. Assume that X is a metric space with compression δ > 0. Choose 0 < p < δ and for all n ∈ N 0 , let R n = n r , ǫ n = 1 an b for some a, b, r ∈ R + . For every n large enough, we can find a collection of unit vectors (ξ x n ) x∈X in some Hilbert space H p satisfying Proof. Let F : X → H be a uniform embedding of X into a Hilbert space satisfying ∀x, y ∈ X : It is easy to verify that the vectors (ξ x n ) x∈X satisfy Condition (1) of this Corollary. Regarding the second condition, note that Consequently we have ξ x n − ξ y n ≥ 1 whenever This is true if and only if , if and only if If n is large enough, then A n ≤ n b+r+p δ−p , so we obtain ξ x n − ξ y n ≥ 1 provided d(x, y) ≥ n r+b+p δ−p .
Remark 2.7. In the second part of the proof of Proposition 2.5, we need the condition to prove that F is Lipschitz. Assume now that X is a finitely generated group which is equipped with the word length metric relative to some finite symmetric generating subset. Then X is a geodesic metric space implying that any function ρ + satisfying ∀x, y ∈ X : F (x) − F (y) ≤ ρ + (d(x, y)), can assumed to be of the form Cd(x, y) + D for some constants C, D ≥ 0. Therefore, in order to prove that F is Lipschitz, we can relax condition (2) to x ′ ) ≥ S n , then holds for smaller S n and we obtain that the function ρ − = 1 2 ∞ n=1 √ n − 1χ [Sn−1,Sn) becomes larger. This will help us to get better compression estimates later.
Proof of Theorem 2.2 Choose n ∈ N 0 , p > 0 and denote R = √ n, ǫ = 1 n 1/2+p . Next, take g(n) ∈ N such that x ∈ G g(n) whenever l G (x) ≤ R = √ n. Set t = − ln(1−ǫ 2 /2) ( C g(n) R+ D g(n) ) 2 and take vectors (ξ x ) x∈G g(n) as in the proof of proposition 2.5, i.e. such that for all x, y ∈ G g(n) : From the lower bound on ξ x , ξ y , one derives Calculating as in Corollary 2.6, we derive that ξ x − ξ y ≥ 1 whenever d g(n) (x, y) ≥ S n := In the proof of Proposition 3.1 of [3], Dadarlat and Guentner explain how the family (ξ x ) x∈G g(n) can be extended to a family of unit vectors (ξ x ) x∈G in a larger Hilbert space, but still satisfying similar inequalities. More precisely, we obtain unit vectors (ξ x ) x∈G in a Hilbert space satisfying From the proof of proposition 2.5, we derive the existence of a large-scale uniform embedding of G into a Hilbert space whose compression function ρ − , is greater than 1 . Choose some β ∈ [0, 1], and define γ : R + → R + , t → t β . If γ eventually lies under some multiple of ρ − , then the compression of G is greater than β. There exists T, C ∈ R + such that if and only if n + D g(n) ) + D g(n) )) }.
Recalling that lim n→∞ [( − ln (2) ln(1− 1 2n 2p+1 ) )/(2 ln(2)n 2p+1 )] = 1 and that we can let p go to 0 since it is just a positive real number that we've chosen, we get the desired lower bound for the Hilbert space compression of G.
Remark 2.8. If G happens to be a quasi-geodesic space, then we can use Remark 2.7 to improve our result. Using the same notations as in Theorem 2.2 and assuming that G is a quasi-geodesic space, we obtain the following. Denote g : N → N a function such that for all x ∈ G we have x ∈ G g(n) whenever l(x) ≤ ln(n). Then, α(G) ≥ lim sup n→∞ (δ/2) ln(n − 1) ln(C g(n) 2 ln(2)n( C g(n) ln(n) + D g(n) ) + C g(n) D g(n) ) . Remark 2.9. All of the above easily generalizes to directed systems of groups (G i ) i∈I where I is any directed set.

Hilbert space compression for group extensions
In this paragraph, Γ will denote a group whose metric is induced by a length function l Γ and H will denote a normal subgroup of Γ which has strictly positive Hilbert space compression when equipped with the induced metric l H := (l Γ ) |H . We assume that Γ is a group extension and we equip G with the induced length from Γ, i.e. for all x ∈ Γ : l G (π(x)) := inf{l Γ (y) | y ∈ Γ, π(y) = π(x)}. Given certain conditions on G, we shall give bounds on the Hilbert space compression of Γ.
3.1. Extensions by a group of polynomial growth. Let us begin by recalling the definition of a metric space with polynomial growth. Definition 3.1. A metric space X has polynomial growth if there exists a polynomial P such that | B(x, R) |≤ P (R) for every x ∈ X and every R ≥ 0. Here B(x, R) is the closed ball with radius R and center x.
Notice that G can have polynomial growth only if l G is proper. In Lemma 6.6 of [12], Tu proves that groups of polynomial growth have property A. We obtain the following lemma by quantifying his proof.
Lemma 3.2. Let G be a group, equipped with a proper length function, that has polynomial growth. Let p ∈]0, 1[ be any real number. There exists n 0 ∈ N such that for every natural number n ≥ n 0 , there exists a collection of unit vectors (g n (x)) x∈G in l 2 (G) such that g n (x) 2 = 1, ∀x ∈ G and Proof. For each x ∈ G and r ∈ R, denote by B(x, r) ⊂ G the ball of radius r and center x. Denote the characteristic function of B(x, r) by χ r x . We shall denote B(1, r) simply by B r and χ r 1 by χ r . For n ∈ N 0 , denote R n = √ n and, with the convention that ∀a ∈ R : a/0 = ∞, let k(n) be the infimum of all real numbers r such that Clearly, such k(n) exists, since if it didn't exist, then ∀i ∈ N 0 , obtaining a contradiction since the left hand side depends polynomially on i whereas the right hand side depends exponentially on i.
We claim that there exists n ∈ N 0 such that ∀n ≥ n : k(n) ≤ 2n 3/2+4p . Assume therefore, that such n does not exist. Then there exists a strictly monotone increasing sequence (n i ) i∈N such that ∀i : Denoting the integer part of a real number a by [a] and assuming for the last inequality below that ∀i : n i ≥ 2 1/p , we obtain that (1/2), it is clear that the right hand side depends exponentially on n i , whereas the left hand side depends polynomially on n i . We obtain a contradiction.
Consider now the functions χ k(n) x . They are elements of l 1 (G) such that d G (x, y) ≤ √ n = R n implies Moreover, the support of χ k(n) x lies inside B(x, k(n)) ⊂ B(x, 2k(n)) ⊂ B(x, 4n 3/2+4p ) ⊂ B(x, n 3/2+5p ) whenever n is larger than some natural number n 0 . To conclude, take n ≥ max(n 0 , n) and define g n (x) = χ k(n) x χ k(n) 1 . Clearly, these are elements of norm 1 in l 2 (G) that satisfy condition (2) of this lemma. To show that they also satisfy condition (1), take x, y such that d G (x, y) ≤ R n . Then Therefore | 1 − g n (x), g n (y) |≤ 1 4n 1+2p as desired.
Theorem 3.3. Assume that Γ is a group, equipped with some length function l = l Γ , that fits in a short exact sequence 1 δ−p . For n sufficiently large, Lemma 3.2 provides maps g n : G → l 2 (G) such that g n (x) 2 = 1, ∀x ∈ G and such that • | 1 − g n (x), g n (y) |≤ 1 4n 1+2p provided d G (x, y) ≤ √ n, • supp(g n (x)) ⊂ B(x, S G n ) for all x ∈ G where S G n = n 3/2+5p . In the proof of Theorem 4.1 in [3], Guentner and Kaminker fix n and use the maps g n and h n to construct a map f n : Γ → l 2 (G, H) such that f n (a) = 1, ∀a ∈ Γ and n + S H n . Denoting S n = n p S H n , we obtain for n larger than some From the second part of the proof of Proposition 2.5, we find a Lipschitz uniform embedding F of Γ into a Hilbert space such that F (x)−F (y) ≥ ρ − (d(x, y)) := 1 2 ∞ n=n1+1 √ n − n 1 χ [Sn−1,Sn[ (d(x, y)) for d(x, y) ∈ [S n1 , ∞[. A function u : R + → R + of the form x → x β for some β > 0 eventually lies under a constant C times ρ − whenever there exists N such that for all n ≥ N, S n β ≤ C √ n − n 1 .
Since p was just any number between 0 and δ, we decide to let p go to 0, obtaining that the compression of Γ is at least δ/4.

3.2.
Extensions by a finitely generated word-hyperbolic group. We start by a lemma similar to Lemma 3.2. More precisely, quantifying the proof of Proposition 8.1 in [12], we obtain Lemma 3.4. Let G be a finitely generated hyperbolic group. Let p ∈]0, 1[ be any real number. There exists n 0 ∈ N such that for every natural number n ≥ n 0 , there exists a collection of unit vectors (g n (x)) x∈G in l 2 (G) such that g n (x) = 1, ∀x ∈ G and • | 1 − g n (x), g n (y) |≤ 1 4n 1+2p provided d G (x, y) ≤ ln(n), • supp(g n (x)) ⊂ B(x, S G n ) for all x ∈ G where S G n = n 2+6p .