Weak Amenability of Hyperbolic Groups
WEAK AMENABILITY OF HYPERBOLIC GROUPS
NARUTAKA OZAWA
Abstract. We prove that hyperbolic groups are weakly amenable. This par-
tially extends the result of Cowling and Haagerup showing that lattices in
simple Lie groups of real rank one are weakly amenable. We take a combina-
torial approach in the spirit of Haagerup and prove that for the word length
distance d of a hyperbolic group, the Schur multipliers associated with the
kernel rd have uniformly bounded norms for 0 < r < 1. We then combine
this with a Bożejko-Picardello type inequality to obtain weak amenability.
1. Introduction
The notion of weak amenability for groups was introduced by Cowling and
Haagerup [CH]. (It has almost nothing to do with the notion of weak amenability
for Banach algebras.) We use the following equivalent form of the definition. See
Section 2 and [BO, CH, Pi] for more information.
Definition. A countable discrete group Γ is said to be weakly amenable with
constant C if there exists a sequence of finitely supported functions ϕn on Γ
such that ϕn → 1 pointwise and supn ‖ϕn‖cb ≤ C, where ‖ϕ‖cb denotes the
(completely bounded) norm of the Schur multiplier on B(ℓ2Γ) associated with
(x, y) 7→ ϕ(x−1y).
In the pioneering paper [Ha], Haagerup proved that the group C∗-algebra of a
free group has a very interesting approximation property. Among other things, he
proved that the graph distance d on a tree Γ is conditionally negatively definite;
in particular, the Schur multiplier on B(ℓ2Γ) associated with the kernel r
d has
(completely bounded) norm one for every 0 < r < 1. For information of Schur
multipliers and completely bounded maps, see Section 2 and [BO, CH, Pi]. Bożejko
and Picardello [BP] proved that the Schur multiplier associated with the charac-
teristic function of the subset {(x, y) : d(x, y) = n} has (completely bounded)
norm at most 2(n + 1). These two results together imply that a group acting
properly on a tree is weakly amenable with constant one. Recently, this result was
extended to the case of finite-dimensional CAT(0) cube complexes by Guentner
and Higson [GH]. See also [Mi]. Cowling and Haagerup [dCH, Co, CH] proved that
lattices in simple Lie groups of real rank one are weakly amenable and computed
explicitly the associated constants. It is then natural to explore this property for
hyperbolic groups in the sense of Gromov [GdH, Gr]. We prove that hyperbolic
2000 Mathematics Subject Classification. Primary 20F67; Secondary 43A65, 46L07.
Key words and phrases. hyperbolic groups, weak amenability, Schur multipliers.
Supported by Sloan Foundation and NSF grant.
http://arxiv.org/abs/0704.1635v3
2 NARUTAKA OZAWA
groups are weakly amenable, without giving estimates of the associated constants.
The results and proofs are inspired by and partially generalize those of Haagerup
[Ha], Pytlik-Szwarc [PS] and Bożejko-Picardello [BP]. We denote by N0 the set of
non-negative integers, and by D the unit disk {z ∈ C : |z| < 1}.
Theorem 1. Let Γ be a hyperbolic graph with bounded degree and d be the graph
distance on Γ. Then, there exists a constant C such that the following are true.
(1) For every z ∈ D, the Schur multiplier θz on B(ℓ2Γ) associated with the
kernel
Γ× Γ ∋ (x, y) 7→ zd(x,y) ∈ C
has (completely bounded) norm at most C|1−z|/(1−|z|). Moreover, z 7→ θz
is a holomorphic map from D into the space V2(Γ) of Schur multipliers.
(2) For every n ∈ N0, the Schur multiplier on B(ℓ2Γ) associated with the
characteristic function of the subset
{(x, y) ∈ Γ× Γ : d(x, y) = n}
has (completely bounded) norm at most C(n+ 1).
(3) There exists a sequence of finitely supported functions fn : N0 → [0, 1] such
that fn → 1 pointwise and that the Schur multiplier on B(ℓ2Γ) associated
with the kernel
Γ× Γ ∋ (x, y) 7→ fn(d(x, y)) ∈ [0, 1]
has (completely bounded) norm at most C for every n.
Let Γ be a hyperbolic group and d be the word length distance associated with
a fixed finite generating subset of Γ. Then, for the sequence fn as above, the
sequence of functions ϕn(x) = fn(d(e, x)) satisfy the properties required for weak
amenability. Thus we obtain the following as a corollary.
Theorem 2. Every hyperbolic group is weakly amenable.
This solves affirmatively a problem raised by Roe at the end of [Ro]. We close
the introduction with a few problems and remarks. Is it possible to construct a
family of uniformly bounded representations as it is done in [Do, PS]? Is it true
that a group which is hyperbolic relative to weakly amenable groups is again weakly
amenable? There is no serious difficulty in extending Theorem 1 to (uniformly)
fine hyperbolic graphs in the sense of Bowditch [Bo]. Ricard and Xu [RX] proved
that weak amenability with constant one is closed under free products with finite
amalgamation. The author is grateful to Professor Masaki Izumi for conversations
and encouragement.
2. Preliminary on Schur multipliers
Let Γ be a set and denote by B(ℓ2Γ) the Banach space of bounded linear oper-
ators on ℓ2Γ. We view an element A ∈ B(ℓ2Γ) as a Γ× Γ-matrix: A = [Ax,y]x,y∈Γ
with Ax,y = 〈Aδy, δx〉. For a kernel k : Γ×Γ → C, the Schur multiplier associated
with k is the map mk on B(ℓ2Γ) defined by mk(A) = [k(x, y)Ax,y]. We recall the
necessary and sufficient condition for mk to be bounded (and everywhere-defined).
See [BO, Pi] for more information of completely bounded maps and the proof of
the following theorem.
WEAK AMENABILITY OF HYPERBOLIC GROUPS 3
Theorem 3. Let a kernel k : Γ × Γ → C and a constant C ≥ 0 be given. Then
the following are equivalent.
(1) The Schur multiplier mk is bounded and ‖mk‖ ≤ C.
(2) The Schur multiplier mk is completely bounded and ‖mk‖cb ≤ C.
(3) There exist a Hilbert space H and vectors ζ+(x), ζ−(y) in H with norms
at most
C such that 〈ζ−(y), ζ+(x)〉 = k(x, y) for every x, y ∈ Γ.
We denote by V2(Γ) = {mk : ‖mk‖ < ∞} the Banach space of Schur multipliers.
The above theorem says that the sesquilinear form
ℓ∞(Γ,H)× ℓ∞(Γ,H) ∋ (ζ−, ζ+) 7→ mk ∈ V2(Γ),
where k(x, y) = 〈ζ−(y), ζ+(x)〉, is contractive for any Hilbert space H.
Let Pf(Γ) be the set of finite subsets of Γ. We note that the empty set ∅ belongs
to Pf (Γ). For S ∈ Pf(Γ), we define ξ̃+S and ξ̃
∈ ℓ2(Pf (Γ)) by
ξ̃+S (ω) =
1 if ω ⊂ S
0 otherwise
and ξ̃−S (ω) =
(−1)|ω| if ω ⊂ S
0 otherwise
We also set ξ+
= ξ̃+
− δ∅ and ξ−S = −(ξ̃
− δ∅). Note that ξ±S ⊥ ξ
if S ∩ T = ∅.
The following lemma is a trivial consequence of the binomial theorem.
Lemma 4. One has ‖ξ±
‖2 + 1 = ‖ξ̃±
‖2 = 2|S| and
〉 = 1− 〈ξ̃−
, ξ̃+
1 if S ∩ T 6= ∅
0 otherwise
for every S, T ∈ Pf(Γ).
3. Preliminary on hyperbolic graphs
We recall and prove some facts of hyperbolic graphs. We identify a graph Γ
with its vertex set and equip it with the graph distance:
d(x, y) = min{n : ∃x = x0, x1, . . . , xn = y such that xi and xi+1 are adjacent}.
We assume the graph Γ to be connected so that d is well-defined. For a subset
E ⊂ Γ and R > 0, we define the R-neighborhood of E by
NR(E) = {x ∈ Γ : d(x,E) < R},
where d(x,E) = inf{d(x, y) : y ∈ E}. We write BR(x) = NR({x}) for the ball
with center x and radius R. A geodesic path p is a finite or infinite sequence of
points in Γ such that d(p(m), p(n)) = |m − n| for every m,n. Most of the time,
we view a geodesic path p as a subset of Γ. We note the following fact (see e.g.,
Lemma E.8 in [BO]).
Lemma 5. Let Γ be a connected graph. Then, for any infinite geodesic path
p : N0 → Γ and any x ∈ Γ, there exists an infinite geodesic path px which starts at
x and eventually flows into p (i.e., the symmetric difference p△ px is finite).
Definition. We say a graph Γ is hyperbolic if there exists a constant δ > 0 such
that for every geodesic triangle each edge is contained in the δ-neighborhood of
the union of the other two. We say a finitely generated group Γ is hyperbolic if its
4 NARUTAKA OZAWA
Cayley graph is hyperbolic. Hyperbolicity is a property of Γ which is independent
of the choice of the finite generating subset [GdH, Gr].
From now on, we consider a hyperbolic graph Γ which has bounded degree:
supx |BR(x)| < ∞ for every R > 0. We fix δ > 1 satisfying the above definition.
We fix once for all an infinite geodesic path p : N0 → Γ and, for every x ∈ Γ,
choose an infinite geodesic path px which starts at x and eventually flows into p.
For x, y, w ∈ Γ, the Gromov product is defined by
〈x, y〉w =
(d(x,w) + d(y, w)− d(x, y)) ≥ 0.
See [BO, GdH, Gr] for more information on hyperbolic spaces and the proof of
the following lemma which says every geodesic triangle is “thin”.
Lemma 6 (Proposition 2.21 in [GdH]). Let x, y, w ∈ Γ be arbitrary. Then, for
any geodesic path [x, y] connecting x to y, one has d(w, [x, y]) ≤ 〈x, y〉w + 10δ.
Lemma 7. For x ∈ Γ and k ∈ Z, we set
T (x, k) = {w ∈ N100δ(px) : d(w, x) ∈ {k − 1, k} },
where T (x, k) = ∅ if k < 0. Then, there exists a constant R0 satisfying the
following: For every x ∈ Γ and k ∈ N0, if we denote by v the point on px such
that d(v, x) = k, then
T (x, k) ⊂ BR0(v).
Proof. Let w ∈ T (x, k) and choose a point w′ on px such that d(w,w′) < 100δ.
Then, one has |d(w′, x)− d(w, x)| < 100δ and
d(w, v) ≤ d(w,w′) + d(w′, v) ≤ 100δ + |d(w′, x)− k| < 200δ + 1.
Thus the assertion holds for R0 = 200δ + 1. �
Lemma 8. For k, l ∈ Z, we set
W (k, l) = {(x, y) ∈ Γ× Γ : T (x, k) ∩ T (y, l) 6= ∅}.
Then, for every n ∈ N0, one has
E(n) := {(x, y) ∈ Γ× Γ : d(x, y) ≤ n} =
W (k, n− k).
Moreover, there exists a constant R1 such that
W (k, l) ∩W (k + j, l − j) = ∅
for all j > R1.
Proof. First, if (x, y) ∈ W (k, n−k), then one can find w ∈ T (x, k)∩T (y, n−k) and
d(x, y) ≤ d(x,w) + d(w, y) ≤ n. This proves that the right hand side is contained
in the left hand side. To prove the other inclusion, let (x, y) and n ≥ d(x, y) be
given. Choose a point p on px ∩ py such that d(p, x) + d(p, y) ≥ n, and a geodesic
path [x, y] connecting x to y. By Lemma 6, there is a point a on [x, y] such that
d(a, p) ≤ 〈x, y〉p + 10δ. It follows that
〈x, p〉a + 〈y, p〉a = d(a, p)− 〈x, y〉p ≤ 10δ.
WEAK AMENABILITY OF HYPERBOLIC GROUPS 5
We choose a geodesic path [a, p] connecting a to p and denote by w(m) the point
on [a, p] such that d(w(m), a) = m. Consider the function f(m) = d(w(m), x) +
d(w(m), y). Then, one has that f(0) = d(x, y) ≤ n ≤ d(p, x) + d(p, y) = f(d(a, p))
and that f(m + 1) ≤ f(m) + 2 for every m. Therefore, there is m0 ∈ N0 such
that f(m0) ∈ {n− 1, n}. We claim that w := w(m0) ∈ T (x, k) ∩ T (y, n− k) for
k = d(w, x). First, note that d(w, y) = f(m0)− k ∈ {n− k − 1, n− k}. Since
〈x, p〉w ≤
(d(x, a) + d(a, w) + d(p, w) − d(x, p))
(d(x, a) + d(p, a)− d(x, p))
= 〈x, p〉a
≤ 10δ,
one has that d(w, px) ≤ 20δ by Lemma 6. This proves that w ∈ T (x, k). One
proves likewise that w ∈ T (y, n − k). Therefore, T (x, k) ∩ T (y, n − k) 6= ∅ and
(x, y) ∈ W (k, n− k).
Suppose now that (x, y) ∈ W (k, l) ∩ W (k + j, l − j) exists. We choose v ∈
T (x, k) ∩ T (y, l) and w ∈ T (x, k + j) ∩ T (y, l− j). Let vx (resp. wx) be the point
on px such that d(vx, x) = k (resp. d(wx, x) = k + j). Then, by Lemma 7, one
has d(v, vx) ≤ R0 and d(w,wx) ≤ R0. We choose vy, wy on py likewise for y. It
follows that d(vx, vy) ≤ 2R0 and d(wx, wy) ≤ 2R0. Choose a point p on px ∩ py.
Then, one has |d(vx, p)− d(vy , p)| ≤ 2R0 and |d(wx, p)− d(wy , p)| ≤ 2R0. On the
other hand, one has d(vx, p) = d(wx, p) + j and d(vy , p) = d(wy , p)− j. It follows
2j = d(vx, p)− d(wx, p)− d(vy, p) + d(wy , p) ≤ 4R0.
This proves the second assertion for R1 = 2R0. �
Lemma 9. We set
Z(k, l) = W (k, l) ∩
W (k + j, l − j)c.
Then, for every n ∈ N0, one has
χE(n) =
χZ(k,n−k).
Proof. We first note that Lemma 8 implies Z(k, l) = W (k, l)∩
j=1 W (k+j, l−j)c
k=0 Z(k, n−k) ⊂
k=0 W (k, n−k) = E(n). It is left to show that for every
(x, y) and n ≥ d(x, y), there exists one and only one k such that (x, y) ∈ Z(k, n−k).
For this, we observe that (x, y) ∈ Z(k, n− k) if and only if k is the largest integer
that satisfies (x, y) ∈ W (k, n− k). �
4. Proof of Theorem
Proposition 10. Let Γ be a hyperbolic graph with bounded degree and define
E(n) = {(x, y) : d(x, y) ≤ n}. Then, there exist a constant C0 > 0, subsets
Z(k, l) ⊂ Γ, a Hilbert space H and vectors η+
(x) and η−
(y) in H which satisfy the
following properties:
6 NARUTAKA OZAWA
(1) η±m(w) ⊥ η±m′(w) for every w ∈ Γ and m,m′ ∈ N0 with |m−m′| ≥ 2.
(2) ‖η±m(w)‖ ≤
C0 for every w ∈ Γ and m ∈ N0.
(3) 〈η−
(y), η+
(x)〉 = χZ(k,l)(x, y) for every x, y ∈ Γ and k, l ∈ N0.
(4) χE(n) =
k=0 χZ(k,n−k) for every n ∈ N0.
Proof. We use the same notations as in the previous sections.
Let H = ℓ2(Pf (Γ))⊗(1+R1) and define η+k (x) and η
(y) in H by
(x) = ξ+
T (x,k)
⊗ ξ̃+
T (x,k+1)
⊗ · · · ⊗ ξ̃+
T (x,k+R1)
(y) = ξ−
T (y,l)
⊗ ξ̃−
T (y,l−1)
⊗ · · · ⊗ ξ̃−
T (y,l−R1)
If |m−m′| ≥ 2, then T (w,m)∩T (w,m′) = ∅ and ξ±
T (w,m)
T (w,m′)
. This implies
the first assertion. By Lemma 7 and the assumption that Γ has bounded degree,
one has C1 := supw,m |T (w,m)| ≤ supv |BR0(v)| < ∞. Now the second assertion
follows from Lemma 4 with C0 = 2
C1(1+R1). Finally, by Lemma 4, one has
(y), η+
(x)〉 = χW (k,l)(x, y)
χW (k+j,l−j)c (x, y) = χZ(k,l)(x, y).
This proves the third assertion. The fourth is nothing but Lemma 9. �
Proof of Theorem 1. Take η±m ∈ ℓ∞(Γ,H) as in Proposition 10 and set C = 2C0.
For every z ∈ D, we define ζ±z ∈ ℓ∞(Γ,H) by the absolutely convergent series
ζ+z (x) =
ζ−z (y) =
where
1− z denotes the principal branch of the square root. The construction
of ζ±z draws upon [PS]. We note that the map D ∋ z 7→ (ζ±z (w))w ∈ ℓ∞(Γ,H) is
(anti-)holomorphic. By Proposition 10, one has
〈ζ−z (y), ζ+z (x)〉 = (1 − z)
zk+lχZ(k,l)(x, y)
= (1 − z)
znχE(n)(x, y)
= (1 − z)
n=d(x,y)
= zd(x,y)
WEAK AMENABILITY OF HYPERBOLIC GROUPS 7
for all x, y ∈ Γ, and
‖ζ±z (w)‖2 ≤ 2|1− z|
j=0,1
(z±)2m+jη±2m+j(w)‖
= 2|1− z|
j=0,1
|z|4m+2j‖η±2m+j(w)‖
≤ 2|1− z| 1
1− |z|2C0
|1− z|
1− |z|
for all w ∈ Γ. Therefore the Schur multiplier θz associated with the kernel zd has
(completely bounded) norm at most C|1 − z|/(1− |z|) by Theorem 3. Moreover,
the map D ∋ z 7→ θz ∈ V2(Γ) is holomorphic.
For the second assertion, we simply write ‖Z‖ for the (completely bounded)
norm of the Schur multiplier associated with the characteristic function χZ of a
subset Z ⊂ Γ× Γ. By Proposition 10 and Theorem 3, one has
‖E(n)‖ ≤
‖Z(k, n− k)‖ ≤ C0(n+ 1).
and ‖{(x, y) : d(x, y) = n}‖ = ‖E(n) \ E(n − 1)‖ ≤ C(n + 1). This proves the
second assertion. The third assertion follows from the previous two, by choosing
fn(d) = χE(Kn)(d)r
n for suitable 0 < rn < 1 and Kn ∈ N0 with rn → 1 and
Kn → ∞. We refer to [BP, Ha] for the proof of this fact. �
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Department of Mathematical Sciences, University of Tokyo, Komaba, 153-8914,
Department of Mathematics, UCLA, Los Angeles, CA 90095-1555
E-mail address: narutaka@ms.u-tokyo.ac.jp
	1. Introduction
	2. Preliminary on Schur multipliers
	3. Preliminary on hyperbolic graphs
	4. Proof of Theorem
	References
