THE BAUM-CONNES CONJECTURE FOR HYPERBOLIC GROUPS

Igor Mineyev and Guoliang Yu^{1}

Abstract. We prove the Baum-Connes conjecture for hyperbolic groups and their subgroups.

1. Introduction.

The Baum-Connes conjecture states that, for a discrete group G, the K-homology groups of the classifying space for proper G-action is isomorphic to the K-groups of the reduced group C^{∗}-algebra of G [3, 2]. A positive answer to the Baum-Connes conjecture would provide a complete solution to the problem of computing higher indices of elliptic operators on compact manifolds. The rational injectivity part of the Baum-Connes conjecture implies the Novikov conjecture on homotopy invariance of higher signatures. The Baum-Connes conjecture also implies the Kadison-Kaplansky conjecture that for G torsion free there exists no non-trivial projection in the reduced group C^{∗}-algebra associated to G. In [7], Higson and Kasparov prove the Baum-Connes conjecture for groups acting properly and isometrically on a Hilbert space. In a recent remarkable work, Vincent Lafforgue proves the Baum-Connes conjecture for strongly bolic groups with property RD [15, 12, 13]. In particular, this implies the Baum-Connes conjecture for the fundamental groups of strictly negatively curved compact manifolds. In [4], Connes and Moscovici prove the rational injectivity part of the Baum-Connes conjecture for hyperbolic groups using cyclic cohomology method. In this paper, we exploit Lafforgue’s work to prove the Baum-Connes conjecture for hyperbolic groups and their subgroups.

The main step in the proof is the following theorem.

Theorem 17. Every hyperbolic group G admits a metric d̂ with the following properties.

(1) d̂ is G-invariant, i.e. d̂(g · x, g · y) = d̂(x, y) for all x, y, g ∈ G.

(2) d̂ is quasiisometric to the word metric.

(3) The metric space (G, d̂) is weakly geodesic and strongly bolic.

This paper is organized as follows. In section 2, we recall the concepts of hyperbolic groups and bicombings. In section 3, we introduce a distance-like function r on a hyperbolic group and study its basic properties. In section 4, we prove that r satisfies certain distance-like inequalities. In section 5, we construct a metric d̂ on a hyperbolic group and prove Theorem 17 stated above. In section 6, we combine Lafforgue’s work and Theorem 17 to prove the Baum-Connes conjecture for hyperbolic groups and their subgroups.

1The second author is partially supported by NSF and MSRI.

After this work was done, we learned from Vincent Lafforgue that he has independently proved the Baum-Connes conjecture for hyperbolic groups by a different and elegant method [14], and we also learned from Michael Puschnigg that he has independently proved the Kadison-Kaplansky conjecture for hyperbolic groups using a beautiful local cyclic homology method [17]. It is our pleasure to thank both of them for bringing their work to our attention.

2. Hyperbolic groups and bicombings.

In this section, we recall the concepts of hyperbolic groups and bicombings.

2.1. Hyperbolic groups. Let G be a finitely generated group. Let S be a finite generating set for G. Recall that the Cayley graph of G with respect to S is the graph Γ satisfying the following conditions:

(1) the set of vertices in Γ, denoted by Γ^{(0)}, is G;

(2) the set of edges is G × S, where each edge (g, s) ∈ G × S spans the vertices g and gs.

We endow Γ with the path metric d induced by assigning length 1 to each edge. Notice that G acts freely, isometrically and cocompactly on Γ. A geodesic path in Γ is a shortest edge path. The restriction of the path metric d to G is called the word metric.

A finitely generated group G is called hyperbolic if there exists a constant δ ≥ 0 such that all the geodesic triangles in Γ are δ-fine in the following sense: if a, b, and c are vertices in Γ, [a, b], [b, c], and [c, a] are geodesics from a to b, from b to c, and from c to a, respectively, and points ā ∈ [b, c], v, c̄ ∈ [a, b], w, b̄ ∈ [a, c] satisfy

d(b, c̄) = d(b, ā), d(c, ā) = d(c, b̄), d(a, v) = d(a, w) ≤ d(a, c̄) = d(a, b̄),

then d(v, w) ≤ δ.

The above definition of hyperbolicity does not depend on the choice of the finite generating set S. See [6, 1] for other equivalent definitions.

For vertices a, b, and c in Γ, the Gromov product is defined by

(b|c)_{a} := d(a, b̄) = d(a, c̄) = \frac{1}{2}[d(a, b) + d(a, c) − d(b, c)].

The Gromov product can be used to measure the degree of cancellation in the multiplication of group elements in G.

2.2. Bicombings. Let G be a finitely generated group. Let Γ be a Cayley graph with respect to a finite generating set. A bicombing p in Γ is a function assigning to each ordered pair (a, b) of vertices in Γ an oriented edge-path p[a, b] from a to b. A bicombing p is called geodesic if each path p[a, b] is geodesic, i.e. a shortest edge path. A bicombing p is G-equivariant if p[g · a, g · b] = g · p[a, b] for each a, b ∈ Γ^{(0)} and each g ∈ G.

3. Definition and properties of r(a, b).

The purpose of this section is to introduce a distance-like function r on a hyperbolic group and study its basic properties.

Let G be a hyperbolic group and Γ be a Cayley graph of G with respect to a finite generating set. We endow Γ with the path metric d, and identify G with Γ^{(0)}, the set of vertices of Γ. Let δ ≥ 1 be a positive integer such that all the geodesic triangles in Γ are δ-fine.

The ball B(x, R) is the set of all vertices at distance at most R from the vertex x. The sphere S(x, R) is the set of all vertices at distance R from the vertex x. Pick an equivariant geodesic bicombing p in Γ. By p[a, b](t) we denote the point on the geodesic path p[a, b] at distance t from a. Recall that C_{0}(G, ℚ) is the space of all 0-chains (in G = Γ^{(0)}) with coefficients in ℚ. Endow C_{0}(G, ℚ) with the ℓ^{1}-norm | · |_{1}. We identify G with the standard basis of C_{0}(G, ℚ). Therefore the left action of G on itself induces a left action on C_{0}(G, ℚ).

First we recall several constructions from [16].

For v, w ∈ G, the flower at w with respect to v is defined to be

F l(v, w) := S(v, d(v, w)) ∩ B(w, δ) ⊆ G.

For each a ∈ G, we define pr_{a} : G → G by:

(1) pr_{a}(a) := a;

(2) if b ≠ a, pr_{a}(b) := p[a, b](t), where t is the largest integral multiple of 10δ which is strictly less than d(a, b).

Now for each pair a, b ∈ G, we define a 0-chain f(a, b) in G inductively on the distance d(a, b) as follows:

(1) if d(a, b) ≤ 10δ, f(a, b) := b;

(2) if d(a, b) > 10δ and d(a, b) is not an integral multiple of 10δ, let f(a, b) := f(a, pr_{a}(b));

(3) if d(a, b) > 10δ and d(a, b) is an integral multiple of 10δ, let

f(a, b) := \frac{1}{#F l(a, b)} ∑_{x∈F l(a,b)} f(a, pr_{a}(x)).

Proposition 1 ([16]). The function f : G × G → C_{0}(G, ℚ) defined above satisfies the following conditions.

(1) For each a, b ∈ G, f(b, a) is a convex combination, i.e. its coefficients are nonnegative and sum up to 1.

(2) If d(a, b) ≥ 10δ, then supp f(b, a) ⊆ B(p[b, a](10δ), δ) ∩ S(b, 10δ).

(3) If d(a, b) ≤ 10δ, then f(b, a) = a.

(4) f is G-equivariant, i.e. f(g · b, g · a) = g · f(b, a) for any g, a, b ∈ G.

(5) There exist constants L ≥ 0 and 0 ≤ λ < 1 such that, for all a, a′, b ∈ G,

|f(b, a) − f(b, a′)|_{1} ≤ L λ^{(a|a′)_{b}}.

[figure: a; b; p[b, a]; 10δ; f(b, a)]

Figure 1. Convex combination f(b, a).

Let ω_{7} be the number of elements in a ball of radius 7δ in G. For each a ∈ G, a 0-chain star(a) is defined by

star(a) := \frac{1}{ω_{7}} ∑_{x∈B(a,7δ)} x.

This extends to a linear operator star : C_{0}(G, ℚ) → C_{0}(G, ℚ). Define the 0-chain f̄(b, a) by f̄(b, a) := star(f(b, a)).

The main reason for introducing f̄ is that f̄ has better cancellation properties than f (compare Proposition 1(5) with Proposition 2(5) and 2(6) below). These cancellation properties play key roles in this paper.

Proposition 2 ([16]). The function f̄ : G × G → C_{0}(G, ℚ) defined above satisfies the following conditions.

(1) For each a, b ∈ G, f̄(b, a) is a convex combination.

(2) If d(a, b) ≥ 10δ, then supp f̄(b, a) ⊆ B(p[b, a](10δ), 8δ).

(3) If d(a, b) ≤ 10δ, then supp f̄(b, a) ⊆ B(a, 7δ).

(4) f̄ is G-equivariant, i.e. f̄(g · b, g · a) = g · f̄(b, a) for any g, a, b ∈ G.

(5) There exist constants L ≥ 0 and 0 ≤ λ < 1 such that, for all a, a′, b ∈ G,

|f̄(b, a) − f̄(b, a′)|_{1} ≤ L λ^{(a|a′)_{b}}.

(6) There exists a constant 0 ≤ λ′ < 1 such that if a, b, b′ ∈ G satisfy (a|b)_{b′} ≤ 10δ and (a|b′)_{b} ≤ 10δ, then |f̄(b, a) − f̄(b′, a)|_{1} ≤ 2λ′.

(7) Let a, b, c ∈ G, γ be a geodesic path from a to b, and let

c ∈ N_{G}(γ, 9δ) := {x ∈ G | d(x, γ) ≤ 9δ}.

Then supp(f̄(c, a)) ⊆ N_{G}(γ, 9δ).

Definition 3. For each pair of vertices a, b ∈ G, a rational number r(a, b) ≥ 0 is defined inductively on d(a, b) as follows.

• r(a, a) := 0.

• If 0 < d(a, b) ≤ 10δ, let r(a, b) := 1.

• If d(a, b) > 10δ, let r(a, b) := r(a, f̄(b, a)) + 1, where r(a, f̄(b, a)) is defined by linearity in the second variable.

The function r is well defined by Proposition 2(2). Also, r(a, b) is well defined when b is a 0-chain, by linearity.

Let ℚ_{≥0} denote the set of all non-negative rational numbers.

Proposition 4. For the function r : G × G → ℚ_{≥0} defined above, there exists N ≥ 0 such that, for all a, b, b′ ∈ G,

|r(a, b) − r(a, b′)| ≤ d(b, b′) + N.

Proof. Up to the G-action, there are only finitely many triples of vertices a, b, b′, satisfying d(a, b) + d(a, b′) ≤ 40δ, hence there exists a uniform bound N′ for the norms

|r(a, b) − r(a, b′)|

for such vertices a, b, b′. Let λ′ be the constant from Proposition 2(6) and pick N large enough so that

N′ ≤ N and λ′ · [27δ + N] ≤ N. (1)

We shall prove the inequality in Proposition 4 by induction on d(a, b) + d(a, b′). If d(a, b) + d(a, b′) ≤ 40δ, then

|r(a, b) − r(a, b′)| ≤ N′ ≤ N ≤ d(b, b′) + N

just by the choices of N′ and N. We assume now that d(a, b) + d(a, b′) > 40δ. Consider the following two cases. \underline{Case 1.} (a|b′)_{b} > 10δ or (a|b)_{b′} > 10δ.

[figure: a; b; b′; p[b, a]; γ; 10δ; v′; v; x]

Figure 2. Proposition 4, Case 1.

Assume, for example, that (a|b′)_{b} > 10δ. Then d(a, b) > 10δ, hence, by definition,

r(a, b) = r(a, f̄(b, a)) + 1.

By Proposition 2(2), we have supp f̄(b, a) ⊆ B(v, 8δ), where v := p[b, a](10δ). Also, (a|b′)_{b} > 10δ implies d(b, b′) > 10δ. Hence there exists a geodesic γ between b and b′, and a vertex v′ on γ with d(b, v′) = d(b, v) = 10δ. Since geodesic triangles are δ-fine, d(v, v′) ≤ δ. For every x ∈ supp f̄(b, a),

d(x, b′) ≤ d(x, v) + d(v, v′) + d(v′, b′)
≤ 8δ + δ + [d(b, b′) − 10δ]
≤ d(b, b′) − 1,
d(a, x) ≤ d(a, v) + d(v, x)
≤ [d(a, b) − 10δ] + 8δ
≤ d(a, b) − 1.

Therefore

d(a, x) + d(a, b′) < d(a, b) + d(a, b′).

Hence the induction hypotheses apply to the vertices a, x, and b′, giving

|r(a, x) − r(a, b′)| ≤ d(x, b′) + N ≤ d(b, b′) − 1 + N. (2)

By Proposition 2(1),

f̄(b, a) = ∑_{x∈B(v,8δ)} α_{x}x

for some non-negative coefficients α_{x} summing up to 1. By the definition of r and inequality (2), we have

|r(a, b) − r(a, b′)|

= |r(a, f̄(b, a)) + 1 − r(a, b′)|

= | ∑_{x∈B(v,8δ)} α_{x}r(a, x) + 1 − r(a, b′)|

≤ | ∑_{x∈B(v,8δ)} α_{x}[r(a, x) − r(a, b′)]| + 1

≤ ∑_{x∈B(v,8δ)} α_{x}|r(a, x) − r(a, b′)| + 1

≤ ∑_{x∈B(v,8δ)} α_{x}(d(b, b′) − 1 + N) + 1

= d(b, b′) + N.

\underline{Case 2.} (a|b′)_{b} ≤ 10δ and (a|b)_{b′} ≤ 10δ.

[figure: a; b; b′; p[b, a]; p[b′, a]; b̄′; b̄; ≤10δ; ≤10δ; w; v′; v; x; x′; ≤δ]

Figure 3. Proposition 4, Case 2.

Since d(a, b) + d(a, b′) > 40δ and d(b, b′) = (a|b′)_{b} + (a|b)_{b′} ≤ 20δ, we have d(a, b) > 10δ and d(a, b′) > 10δ. Then, by the definition of r,

|r(a, b) − r(a, b′)| (3)

= |r(a, f̄(b, a)) + 1 − r(a, f̄(b′, a)) − 1|
= |r(a, f̄(b, a) − f̄(b′, a))|.

The 0-chain f̄(b, a) − f̄(b′, a) can be represented in the form f_{+} − f_{−}, where f_{+} and f_{−} are 0-chains with non-negative coefficients and disjoint supports. By Proposition 2(6),

|f_{+}|_{1} + |f_{−}|_{1} = |f_{+} − f_{−}|_{1}
= |f̄(b, a) − f̄(b′, a)|_{1}
≤ 2λ′.

Since the coefficients of the 0-chain f_{+} − f_{−} = f̄(b, a) − f̄(b′, a) sum up to 0, then

|f_{+}|_{1} = |f_{−}|_{1} ≤ λ′. (4)

With the notations v := p[b, a](10δ), v′ := p[b′, a](10δ), we have

supp f_{+} ⊆ supp f̄(b, a) ⊆ B(v, 8δ) and
supp f_{−} ⊆ supp f̄(b′, a) ⊆ B(v′, 8δ)

(see Figure 3). Since geodesic triangles are δ-fine, there exists a point w on p[b, a] such that d(a, w) = d(a, v′) and d(w, v′) ≤ δ. We first assume that d(a, w) ≤ d(a, v). We have

d(v, v′) ≤ d(v, w) + d(w, v′)
≤ d(w, \overline{b′}) + δ
= d(v′, \overline{b}) + δ
≤ 11δ,

where \overline{b′} and \overline{b} are inscribed points as in the definition of δ-fine triangle in section 2.1. If d(a, w) > d(a, v), we can apply the same argument to prove d(v, v′) ≤ 11δ by interchanging v′ with v.

Hence by Proposition 2(2), for each x ∈ supp f_{+} and x′ ∈ supp f_{−},

d(x, x′) ≤ d(x, v) + d(v, v′) + d(v′, x′)
≤ 8δ + 11δ + 8δ
= 27δ.

Also d(a, x) + d(a, x′) < d(a, b) + d(a, b′), so the induction hypotheses for the vertices a, x, and x′ apply, giving

(5)  |r(a, x) − r(a, x′)| ≤ d(x, x′) + N
≤ 27δ + N

for each x ∈ supp f_{+} and x′ ∈ supp f_{−}. Then we continue equality (3) using (4), (5), linearity of r in the second variable, and the definition of N in (1):

|r(a, b) − r(a, b′)| = |r(a, f̄(b, a) − f̄(b′, a))|
= |r(a, f_{+}) − r(a, f_{−})|
≤ λ′ · [27δ + N]
≤ N ≤ d(b, b′) + N.

Proposition 4 is proved.

Let ε : C_{0}(G, ℚ) → ℚ be the augmentation map taking each 0-chain to the sum of its coefficients. A 0-chain z with ε(z) = 0 is called a 0-cycle.

Proposition 5. There exists a constant D ≥ 0 such that, for each a ∈ G and each 0-cycle z,

|r(a, z)| ≤ D |z|_{1} diam(supp(z)).

Proof. It suffices to consider the case z = b−b′, where b and b′ are vertices with d(b, b′) = 1. But this case is immediate from Proposition 4 by taking D := \frac{1}{2}(1 + N).

Theorem 6. For a hyperbolic group G, the function r : G × G → ℚ_{≥0} from Definition 3 satisfies the following properties.

(1) r is G-equivariant, i.e. r(a, b) = r(g · a, g · b) for g, a, b ∈ G.

(2) r is Lipschitz equivalent to the distance function. More precisely, we have

\frac{1}{10δ} d(a, b) ≤ r(a, b) ≤ d(a, b)

for all a, b ∈ G.

(3) There exist constants C ≥ 0 and 0 ≤ μ < 1 such that, for all a, a′, b, b′ ∈ G with d(a, a′) ≤ 1 and d(b, b′) ≤ 1,

|r(a, b) − r(a′, b) − r(a, b′) + r(a′, b′)| ≤ Cμ^{d(a,b)}.

In particular, if d(a, a′) ≤ 1 and d(b, b′) ≤ 1, then

|r(a, b) − r(a′, b) − r(a, b′) + r(a′, b′)| → 0 as d(a, b) → ∞.

Proof. (1) The G-equivariance of r follows from the definition of r and Proposition 2(4).

(2) Using the assumption that δ ≥ 1 and the definition of r, the inequalities

\frac{1}{10δ} d(a, b) ≤ r(a, b) ≤ d(a, b)

can be shown by an easy induction on d(a, b). The remaining part (3) immediately follows from the following proposition.

Proposition 7. There exist constants A > 0, B > 0, and 0 < ρ < 1 such that, for all a, a′, b, b′ ∈ G with d(a, a′) ≤ 1 and d(b, b′) ≤ 30δ,

|r(a, b) − r(a′, b) − r(a, b′) + r(a′, b′)| ≤ (A d(b, b′) + B) ρ^{d(a,b)+d(a,b′)}.

Proof. Let D ≥ 0 be the constant from Proposition 5, L ≥ 0 and 0 ≤ λ < 1 be the constants from Propositions 1(5) and 2(5), δ ≥ 1 be an integral hyperbolicity (finetriangles) constant, and ω_{7} be the number of vertices in a ball of radius 7δ in Γ.

Now we define constants A, B and ρ. Since the inequality obviously holds when b = b′, we will assume that d(b, b′) ≥ 1. Then constant A > 0 can be chosen large enough so that

• the desired inequality is satisfied whenever d(a, b) + d(a, b′) ≤ 100δ, ρ ≥ \sqrt{λ}, and B > 0, and

• 32DδL(\sqrt{λ})^{−32δ} < A.

So from now on we can assume that d(a, b) + d(a, b′) > 100δ. Also the choice of A implies that inequalities

1 − \frac{A}{Al + B} + \frac{32DδL(\sqrt{λ})^{t−32δ}}{(Al + B)ρ^{t−18δ}} ≤ 1 − \frac{A}{Al + B} + \frac{32DδL(\sqrt{λ})^{−32δ}}{Al + B} < 1

hold for all B > 0, \sqrt{λ} ≤ ρ < 1, 1 ≤ l ≤ 30δ, and t ≥ 0. Therefore, we can pick B > 0 sufficiently large and ρ < 1 sufficiently close to 1 so that the inequalities

1 − \frac{A}{Al + B} + \frac{32DδL(\sqrt{λ})^{t−32δ}}{(Al + B)ρ^{t−18δ}} ≤ ρ^{18δ} and

(1 − \frac{1}{ω_{7}} )\frac{30δA + B}{B} + \frac{64DδL(\sqrt{λ})^{t−32δ}}{Bρ^{t−36δ}} ≤ ρ^{36δ}

are satisfied for all 1 ≤ l ≤ 30δ and all t ≥ 0. The above inequalities rewrite as

(A(l − 1) + B) ρ^{t−18δ} + 32DδL (\sqrt{λ})^{t−32δ} ≤ (Al + B) ρ^{t} and (6)

(1 − \frac{1}{ω_{7}} )(30δA + B) ρ^{t−36δ} + 64DδL (\sqrt{λ})^{t−32δ} ≤ B ρ^{t} (7)

and they are satisfied for all 1 ≤ l ≤ 30δ and all t ≥ 0.

The proof of the proposition proceeds by induction on d(a, b) + d(a, b′). We consider the following two cases. \underline{Case 1}. (a|b)_{b′} > 10δ or (a|b′)_{b} > 10δ.

[figure: a; a′; b; b′; p[b, a]; p[b, a′]; 10δ; x]

Figure 4. Proposition 7, Case 1.

Without loss of generality, (a|b′)_{b} > 10δ (interchange b and b′ otherwise). The 0-cycle f(b, a) − f(b, a′) can be uniquely represented as f_{+} − f_{−}, where f_{+} and f_{−} are 0-chains with non-negative coefficients, disjoint supports, and of the same ℓ^{1}-norm. We have

f(b, a) = f_{0} + f_{+} and f(b, a′) = f_{0} + f_{−}

for some 0-chain f_{0} with non-negative coefficients (actually f_{0} = min {f(b, a), f(b, a′)}). Denote α := |f_{+}|_{1} = |f_{−}|_{1} = ε(f_{+}) = ε(f_{−}), where ε is the augmentation map. Since d(a, a′) ≤ 1, then

(a|a′)_{b} ≥ \frac{1}{2} [d(a, b) + d(a′, b) − 1]

≥ \frac{1}{2} [d(a, b) + d(a, b′) − 32δ],

and by Proposition 1(5),

(8)  α = \frac{1}{2} |f(b, a) − f(b, a′)|_{1}

≤ \frac{1}{2}Lλ^{(a|a′)_{b}}

≤ \frac{1}{2}L (\sqrt{λ})^{d(a,b)+d(a,b′)−32δ} .

By the definition of hyperbolicity in section 2.1 and the assumption (a|b′)_{b} > 10δ, we have

d(p[b, a](10δ), p[b, a′](10δ)) ≤ δ.

Hence there exists a vertex x_{0} ∈ B(p[b, a](10δ), 8δ) ∩ B(p[b, a′](10δ), 8δ). By the definitions of r and f̄,

|r(a, b) − r(a′, b) − r(a, b′) + r(a′, b′)|
= |r(a, f̄(b, a)) + 1 − r(a′, f̄(b, a′)) − 1 − r(a, b′) + r(a′, b′)|
= |r(a, star(f_{0} + f_{+})) − r(a′, star(f_{0} + f_{−})) − r(a, b′) + r(a′, b′)|
≤ |r(a, star(f_{0}) + αx_{0}) − r(a′, star(f_{0}) + αx_{0}) − r(a, b′) + r(a′, b′)| +
+|r(a, star(f_{+}) − αx_{0})| + |r(a′, αx_{0} − star(f_{−}))|.

Now we bound each of the three terms in the last sum. We number these terms consecutively as T_{1}, T_{2}, T_{3}. \underline{Term T_{1}}. Using the same argument as in Case 1 in the proof of Proposition 5, one checks that, for each

x ∈ supp(star(f_{0}) + αx_{0}) ⊆ B(p[b, a](10δ), 8δ) ∩ B(p[b, a′](10δ), 8δ),

the following conditions hold:

d(x, b′) ≤ d(b, b′) − 1 ≤ 30δ and
d(a, b) + d(a, b′) − 18δ ≤ d(a, x) + d(a, b′) ≤ d(a, b) + d(a, b′) − 1.

In particular, the induction hypotheses are satisfied for the vertices a, a′, x, b′, giving

|r(a, x) − r(a′, x) − r(a, b′) + r(a′, b′)|
≤ (A d(x, b′) + B) ρ^{d(a,x)+d(a,b′)}
≤ (A(d(b, b′) − 1) + B) ρ^{d(a,b)+d(a,b′)−18δ}.

Since star(f_{0}) + αx_{0} is a convex combination, by linearity of r in the second variable,

T_{1} = |r(a, star(f_{0}) + αx_{0}) − r(a′, star(f_{0}) + αx_{0}) − r(a, b′) + r(a′, b′)|
≤ (A (d(b, b′) − 1) + B) ρ^{d(a,b)+d(a,b′)−18δ}.

12 THE BAUM-CONNES CONJECTURE FOR HYPERBOLIC GROUPS

\underline{Terms T_{2} and T_{3}}. Since star(f_{+}) − αx_{0} is a 0-cycle supported in a ball of radius 8δ, by Proposition 5 and inequality (8),

T_{2} = |r(a, star(f_{+}) − αx_{0})|
≤ D |star(f_{+}) − αx_{0}|_{1} · 16δ
≤ D · 2α · 16δ
≤ 16DδL (\sqrt{λ})^{d(a,b)+d(a,b′)−32δ} .

Analogously,

T_{3} = |r(a′, αx_{0} − star(f_{−}))| ≤ 16DδL (\sqrt{λ})^{d(a,b)+d(a,b′)−32δ} .

Combining the three bounds above and using the definition of B and ρ (inequality (6)), |r(a, b) − r(a′, b) − r(a, b′) + r(a′, b′)|

≤ T_{1} + T_{2} + T_{3}
≤ (A (d(b, b′) − 1) + B) ρ^{d(a,b)+d(a,b′)−18δ} + 32DδL (\sqrt{λ})^{d(a,b)+d(a,b′)−32δ}
≤ (A d(b, b′) + B) ρ^{d(a,b)+d(a,b′)}.

This finishes Case 1. \underline{Case 2}. (a|b)_{b′} ≤ 10δ and (a|b′)_{b} ≤ 10δ.

[figure: a; a′; b; b′; 10δ ≥; ≤10δ; v; v′; x; x′; ≤δ]

Figure 5. Proposition 7, Case 2.

As in Case 1, we have

f(b, a) − f(b, a′) = f_{+} − f_{−}, f(b, a) = f_{0} + f_{+}, f(b, a′) = f_{0} + f_{−}, α := |f_{+}|_{1} = |f_{−}|_{1} = ε(f_{+}) = ε(f_{−}),

α ≤ \frac{1}{2}Lλ^{(a|a′)_{b}} ≤ \frac{1}{2}L (\sqrt{λ})^{d(a,b)+d(a,b′)−32δ} ,

where f_{+}, f_{−}, and f_{0} are 0-chains with non-negative coefficients, and f_{+} and f_{−} have disjoint supports. Analogously, interchanging b and b′,

f(b′, a) − f(b′, a′) = f′_{+} − f′_{−}, f(b′, a) = f′_{0} + f′_{+}, f(b′, a′) = f′_{0} + f′_{−},

α′ := |f′_{+}|_{1} = |f′_{−}|_{1} = ε(f′_{+}) = ε(f′_{−}),

α′ ≤ \frac{1}{2}Lλ^{(a|a′)_{b}} ≤ \frac{1}{2}L (\sqrt{λ})^{d(a,b)+d(a,b′)−32δ} ,

where f′_{+}, f′_{−}, and f′_{0} are 0-chains with non-negative coefficients, and f′_{+} and f′_{−} have disjoint supports.

Denote v := p[b, a](10δ) and v′ := p[b′, a](10δ). By the conditions of Case 2 and δhyperbolicity of Γ, using the same argument as in Case 2 in the proof of Proposition 4, we obtain d(v, v′) ≤ 11δ. Let x_{0} be a vertex closest to the mid-point of a geodesic path connecting v to v′. Proposition 1(2) implies that

supp f_{0} ∪ supp f′_{0} ⊆ B(x_{0}, 7δ) and
supp f_{−} ∪ supp f_{+} ∪ supp f′_{−} ∪ supp f′_{+} ⊆ B(x_{0}, 8δ).

By the definition of r,

|r(a, b) − r(a′, b) − r(a, b′) + r(a′, b′)|
= |r(a, f̄(b, a)) − r(a′, f̄(b, a′)) − r(a, f̄(b′, a)) + r(a′, f̄(b′, a′))|

≤ |r(a, star(f_{0} + f_{+})) − r(a′, star(f_{0} + f_{−})) − r(a, star(f′_{0} + f′_{+})) + r(a′, star(f′_{0} + f′_{−}))|

≤ |r(a, star(f_{0}) + αx_{0} − star(f′_{0}) − α′x_{0}) − r(a′, star(f_{0}) + αx_{0} − star(f′_{0}) − α′x_{0})| +

+|r(a, star(f_{+})) − r(a, αx_{0})| + |r(a′, αx_{0}) − r(a′, star(f_{−}))| +
+|r(a, α′x_{0}) − r(a, star(f′_{+}))| + |r(a′, star(f′_{−})) − r(a′, α′x_{0})|.

Now we bound each of the five terms in the last sum. We number these terms consecutively as S_{1}, ..., S_{5}. \underline{Term S_{1}}. One checks that, for each

x ∈ supp(star(f_{0}) + αx_{0}) ⊆ B(v, 8δ) ∩ B(p[b, a′](10δ), 8δ) and
x′ ∈ supp(star(f′_{0}) + α′x_{0}) ⊆ B(v′, 8δ) ∩ B(p[b′, a′](10δ), 8δ),

the following conditions hold:

d(x, x′) ≤ 30δ and
d(a, b) + d(a, b′) − 36δ ≤ d(a, x) + d(a, x′) ≤ d(a, b) + d(a, b′) − 1.

In particular, the induction hypotheses are satisfied for the vertices a, a′, x, x′, giving

|r(a, x) − r(a′, x) − r(a, x′) + r(a′, x′)| (9)

≤ (A d(x, x′) + B) ρ^{d(a,x)+d(a,x′)}
≤ (30δA + B) ρ^{d(a,b)+d(a,b′)−36δ}.

Recall that ω_{7} is the number of vertices in a ball of radius 7δ. Let β be the (positive) coefficient of x_{0} in the 0-chain star(f_{0}), and β′ be the (positive) coefficient of x_{0} in the 0-chain star(f′_{0}). Without loss of generality, we can assume |star(f_{0})|_{1} ≤ |star(f′_{0})|_{1}. Since x_{0} was chosen so that supp f_{0} ∪ supp f′_{0} ⊆ B(x_{0}, 7δ), by the definition of star, we have

β = \frac{1}{ω_{7}} |f_{0}|_{1} = \frac{1}{ω_{7}} |star(f_{0})|_{1} ≤ \frac{1}{ω_{7}} |star(f′_{0})|_{1} = \frac{1}{ω_{7}} |f′_{0}|_{1} = β′ and

α − α′ = (1 − |f_{0}|_{1}) − (1 − |f′_{0}|_{1}) = |f′_{0}|_{1} − |f_{0}|_{1} = ω_{7}(β′ − β) ≥ 0.

Therefore,

|star(f_{0}) + αx_{0} − star(f′_{0}) − α′x_{0}|_{1}

≤ |star(f_{0}) − βx_{0}|_{1} + |β′x_{0} − star(f′_{0})|_{1} + |[(α − α′) − (β′ − β)]x_{0}|_{1}

= (|star(f_{0})|_{1} − β) + (|star(f′_{0})|_{1} − β′) + (β′ − β)(ω_{7} − 1)

= (|f_{0}|_{1} − β) + (|f′_{0}|_{1} − β′) + (|f′_{0}|_{1} − |f_{0}|_{1})(1 − \frac{1}{ω_{7}} )

= |f_{0}|_{1}(1 − \frac{1}{ω_{7}} ) + |f′_{0}|_{1}(1 − \frac{1}{ω_{7}} ) + (|f′_{0}|_{1} − |f_{0}|_{1})(1 − \frac{1}{ω_{7}} )

= 2|f′_{0}|_{1}(1 − \frac{1}{ω_{7}} )

≤ 2(1 − \frac{1}{ω_{7}} ).

Since [star(f_{0}) + αx_{0}] − [star(f′_{0}) + α′x_{0}] is a 0-cycle, it is of the form h_{+} − h_{−}, where h_{+} and h_{−} are 0-chains with non-negative coefficients, disjoint supports and of the same ℓ^{1}-norm, so we can define

γ := |h_{+}|_{1} = |h_{−}|_{1} = ε(h_{+}) = ε(h_{−}).

By the above inequality,

γ = \frac{1}{2} |h_{+} − h_{−}|_{1} ≤ 1 − \frac{1}{ω_{7}} ,

then, by (9) and linearity of r in the second variable,

S_{1} = |r(a, h_{+} − h_{−}) − r(a′, h_{+} − h_{−})|
= |r(a, h_{+}) − r(a′, h_{+}) − r(a, h_{−}) + r(a′, h_{−})|
≤ γ · (30δA + B) ρ^{d(a,b)+d(a,b′)−36δ}
≤ (1 − \frac{1}{ω_{7}} )(30δA + B) ρ^{d(a,b)+d(a,b′)−36δ}.

\underline{Terms S_{2} − S_{5}}. Analogously to term T_{2} in Case 1,

S_{2} ≤ 16DδL (\sqrt{λ})^{d(a,b)+d(a,b′)−32δ} , S_{3} ≤ 16DδL (\sqrt{λ})^{d(a,b)+d(a,b′)−32δ} ,
S_{4} ≤ 16DδL (\sqrt{λ})^{d(a,b)+d(a,b′)−32δ} , S_{5} ≤ 16DδL (\sqrt{λ})^{d(a,b)+d(a,b′)−32δ} .

Combining the bounds for the five terms above and using the definition of B and ρ (inequality (7)),

|r(a, b) − r(a′, b) − r(a, b′) + r(a′, b′)|

≤ S_{1} + S_{2} + S_{3} + S_{4} + S_{5}

≤ (1 − \frac{1}{ω_{7}} )(30δA + B) ρ^{d(a,b)+d(a,b′)−36δ} + 64DδL (\sqrt{λ})^{d(a,b)+d(a,b′)−32δ}

≤ B ρ^{d(a,b)+d(a,b′)}

≤ (A d(b, b′) + B) ρ^{d(a,b)+d(a,b′)}.

Proposition 7 and Theorem 6 are proved.

4. More properties of r.

In this section, we prove two distance-like inequalities for the function r introduced in the previous section.

As before, let G be a hyperbolic group and Γ be the Cayley graph of G with respect to a finite generating set. For any subset A ⊆ Γ, denote

N_{G}(A, R) := {x ∈ G | d(x, A) ≤ R}.

Proposition 8. There exists C_{1} ≥ 0 with the following property. If a, b ∈ G, γ is a geodesic in Γ connecting a and b, x ∈ G ∩ γ, γ′ is the part of γ between x and b, and c ∈ N_{G}(γ′, 9δ), then

|r(a, c) − r(a, x) − r(x, c)| ≤ C_{1} (F igure 6).

[figure: a; b; c; c′; x; γ]

Figure 6. Proposition 8.

Proof. Let

C_{1} := (80δ + N + 36δDL) ∑_{k=0}^{∞} λ^{k−18δ},

where L ≥ 1 and 0 < λ < 1 are as in Propositions 1(5) and 2(5), N is as in Proposition 4, and D is as in Proposition 5. It suffices to show the inequality

|r(a, c) − r(a, x) − r(x, c)| ≤ (80δ + N + 36δDL) ∑_{k=0}^{d(x,c)} λ^{k−18δ}.

We will prove it by induction on d(x, c).

If d(x, c) ≤ 40δ, by Proposition 4 and Theorem 6(2) we have

|r(a, c) − r(a, x) − r(x, c)| ≤ |r(a, c) − r(a, x)| + r(x, c)

≤ (d(c, x) + N) + d(x, c) ≤ 80δ + N

≤ (80δ + N + 36δDL) ∑_{k=0}^{d(x,c)} λ^{k−18δ}.

Now we assume that d(x, c) > 40δ. There exists a vertex c′ ∈ γ′ with d(c′, c) ≤ 9δ, so

d(a, c) ≥ d(a, c′) − 9δ ≥ d(x, c′) − 9δ ≥ d(x, c) − 18δ > 10δ.

Hence by the definition of the function r, we have

r(a, c) = r(a, f̄(c, a)) + 1 and r(x, c) = r(x, f̄(c, x)) + 1.

Also

(a|x)_{c} = \frac{1}{2}[d(c, a) + d(c, x) − d(a, x)]

≥ \frac{1}{2}[d(c′, a) − 9δ + d(c′, x) − 9δ − d(a, x)]

= d(x, c′) − 9δ ≥ d(x, c) − 18δ.

By Proposition 2(5),

|f̄(c, x) − f̄(c, a)|_{1} ≤ Lλ^{(a|x)_{c}} ≤ Lλ^{d(x,c)−18δ}.

This, together with Proposition 5 and Proposition 2(2), implies that

|r(a, f̄(c, a)) − r(a, f̄(c, x))| = |r(a, f̄(c, a) − f̄(c, x))|
≤ DLλ^{d(x,c)−18δ} diam(supp(f̄(c, a) − f̄(c, x)))
≤ 36δDLλ^{d(x,c)−18δ}.

By Proposition 2(2) and 2(7), for every y ∈ supp(f̄(c, x)), we have

d(x, y) ≤ d(x, c) − 1 and y ∈ N_{G}(γ′, 9δ).

Hence by the induction hypotheses, we obtain

|r(a, c) − r(a, x) − r(x, c)|

= |(r(a, f̄(c, a)) + 1) − r(a, x) − (r(x, f̄(c, x)) + 1)|

≤ |r(a, f̄(c, a)) − r(a, f̄(c, x))| + |r(a, f̄(c, x)) − r(a, x) − r(x, f̄(c, x))|

≤ 36δDLλ^{d(x,c)−18δ} + (80δ + N + 36δDL) ∑_{k=0}^{d(x,c)−1} λ^{k−18δ}

≤ (80δ + N + 36δDL) ∑_{k=0}^{d(x,c)} λ^{k−18δ}.

Proposition 9. There exists M′ ≥ 0 such that

|r(a, b) − r(a′, b)| ≤ M′ d(a, a′)

for all a, a′, b ∈ G.

Proof. Recall that δ ≥ 1. Let

M′ := (20δ + 3 + 36δDL) ∑_{k=0}^{∞} λ^{k−19δ}.

The Cayley graph Γ is a geodesic metric space, hence it suffices to show the inequality |r(a, b) − r(a′, b)| ≤ M′ in the case when d(a, a′) = 1. For that, it suffices to show the inequality

|r(a, b) − r(a′, b)| ≤ (20δ + 3 + 36δDL) ∑_{k=0}^{d(b,a)} λ^{k−19δ}

when d(a, a′) = 1. We will prove it by induction on d(a, b).

If d(a, b) ≤ 10δ + 1, then by Theorem 6(2) we have

|r(a, b) − r(a′, b)| ≤ r(a, b) + r(a′, b)

≤ d(a, b) + d(a′, b)

≤ 20δ + 3

≤ (20δ + 3 + 36δDL) ∑_{k=0}^{d(b,a)} λ^{k−19δ}.

If d(a, b) > 10δ + 1, then d(a′, b) > 10δ.

For every y ∈ supp(f̄(b, a)) ∪ supp(f̄(b, a′)), by Proposition 2(2) we have

(a|a′)_{y} = \frac{1}{2}[d(y, a) + d(y, a′) − d(a, a′)] ≥ d(b, a) − 19δ.

Hence by the definition of the function r, the induction hypothesis and Propositions 2(5) and 5, we obtain

|r(a, b) − r(a′, b)|

= |(r(a, f̄(b, a)) + 1) − (r(a′, f̄(b, a′)) + 1)|

≤ |r(a, f̄(b, a)) − r(a′, f̄(b, a))| + |r(a′, f̄(b, a)) − r(a′, f̄(b, a′))|

≤ (20δ + 3 + 36δDL) ∑_{k=0}^{d(b,a)−1} λ^{k−19δ} + DLλ^{d(b,a)−19δ}diam(supp(f̄(b, a) − f̄(b, a′)))

≤ (20δ + 3 + 36δDL) ∑_{k=0}^{d(b,a)−1} λ^{k−19δ} + 36δDLλ^{d(b,a)−19δ}

≤ (20δ + 3 + 36δDL) ∑_{k=0}^{d(b,a)} λ^{k−19δ}.

5. Definition and properties of a new metric d̂.

In this section, we use the function r defined in section 3 to construct a G-invariant metric d̂ on a hyperbolic group G such that d̂ is quasi-isometric to the word metric and prove that (G, d̂) is weakly geodesic and strongly bolic.

We define

s(a, b) := \frac{1}{2} [r(a, b) + r(b, a)]

for all a, b ∈ G.

Proposition 10. The above function s satisfies the following conditions.

(a) There exists M ≥ 0 such that

|s(u, v) − s(u, v′)| ≤ M d(v, v′) and |s(u, v) − s(u′, v)| ≤ M d(u, u′)

for all u, u′, v, v′ ∈ G.

(b) There exists C_{1} ≥ 0 such that if a vertex w lies on a geodesic connecting vertices u

and v, then

|s(u, v) − s(u, w) − s(w, v)| ≤ C_{1}.

Proof. (a) Since s is symmetric, it suffices to show only the first inequality. Since the Cayley graph Γ is a geodesic metric space, it suffices to consider only the case d(v, v′) = 1. This case follows from Propositions 4 and 9.

(b) follows from Proposition 8.

Proposition 11. There exists C_{2} ≥ 0 such that

s(a, b) ≤ s(a, c) + s(c, b) + C_{2}

for all a, b, c ∈ G.

Proof. Let ā ∈ p[b, c], c̄ ∈ p[a, b], b̄ ∈ p[a, c] such that

d(b, c̄) = d(b, ā), d(c, ā) = d(c, b̄), d(a, c̄) = d(a, b̄).

By the definition of hyperbolicity, we have

d(ā, b̄) ≤ δ, d(ā, c̄) ≤ δ, d(b̄, c̄) ≤ δ.

By Proposition 10,

s(a, b) ≤ s(a, c̄) + s(c̄, b) + C_{1}
≤ (s(a, b̄) + M d(b̄, c̄)) + (s(ā, b) + M d(c̄, ā)) + C_{1}
≤ s(a, b̄) + s(ā, b) + 2δM + C_{1}
≤ (s(a, b̄) + s(b̄, c)) + (s(c, ā) + s(ā, b)) + 2δM + C_{1}
≤ s(a, c) + s(c, b) + 2δM + 3C_{1},

so we set C_{2} := 2δM + 3C_{1}.

For every pair of elements a, b ∈ G, we define

d̂(a, b) := \{ \begin{matrix}s(a, b) + C_{2} & if a ≠ b, \\ 0 & if a = b. \end{matrix}

Proposition 12. The function d̂ defined above is a metric on G.

Proof. By definition, d̂ is symmetric, and d̂(a, b) = 0 iff a = b. The triangle inequality is a direct consequence of Proposition 11.

Proposition 13. There exist constants C ≥ 0 and 0 ≤ μ < 1 with the following property. For all R ≥ 0 and all a, a′, b, b′ ∈ G with d(a, a′) ≤ R and d(b, b′) ≤ R,

|d̂(a, b) − d̂(a′, b) − d̂(a, b′) + d̂(a′, b′)| ≤ R^{2}Cμ^{d(a,b)−2R}.

In particular, if d(a, a′) ≤ R and d(b, b′) ≤ R, then

d̂(a, b) − d̂(a′, b) − d̂(a, b′) + d̂(a′, b′) → 0 as d(a, b) → ∞.

Proof. Take C and μ as in Theorem 6(3). Increasing C if needed we can assume that a ≠ b, a ≠ b′, a′ ≠ b, a′ ≠ b′.

If a = a′ or b = b′, then

d̂(a, b) − d̂(a′, b) − d̂(a, b′) + d̂(a′, b′) = 0.

If d(a, a′) = 1 and d(b, b′) = 1, then by Theorem 6(3),

|d̂(a, b) − d̂(a′, b) − d̂(a, b′) + d̂(a′, b′)|
= |s(a, b) − s(a′, b) − s(a, b′) + s(a′, b′)|
≤ Cμ^{d(a,b)}.

Without loss of generality, we can assume that R is an integer. In the general case

d(a, a′) ≤ R and d(b, b′) ≤ R,

pick vertices a = a_{0}, a_{1}, ..., a_{R} = a′ with d(a_{i−1}, a_{i}) ≤ 1 and b = b_{0}, b_{1}, ..., b_{R} = b′ with d(b_{j−1}, b_{j}) ≤ 1 and note that d(a_{i}, b_{j}) ≥ d(a, b) − 2R. Then we have

|d̂(a, b) − d̂(a′, b) − d̂(a, b′) + d̂(a′, b′)|

= |s(a, b) − s(a′, b) − s(a, b′) + s(a′, b′)|

= | ∑_{i=1}^{R} ∑_{j=1}^{R} (s(a_{i−1}, b_{j−1}) − s(a_{i}, b_{j−1}) − s(a_{i−1}, b_{j}) + s(a_{i}, b_{j}))|

≤ ∑_{i=1}^{R} ∑_{j=1}^{R} |s(a_{i−1}, b_{j−1}) − s(a_{i}, b_{j−1}) − s(a_{i−1}, b_{j}) + s(a_{i}, b_{j})|

≤ R^{2}Cμ^{d(a,b)−2R}.

Recall that a metric space (X, d) is said to be weakly geodesic [11, 10] if there exists δ_{1} ≥ 0 such that, for every pair of points x and y in X and every t ∈ [0, d(x, y)], there exists a point a ∈ X such that d(a, x) ≤ t + δ_{1} and d(a, y) ≤ d(x, y) − t + δ_{1}.

Proposition 14. The metric space (G, d̂) is weakly geodesic.

Proof. Let x, y ∈ G and z ∈ G ∩ p[x, y]. By the definition of d̂ and Proposition 10(b), we have

d̂(x, z) + d̂(z, y) − d̂(x, y) ≤ C_{1} + 2C_{2}.

It follows that

d̂(x, z) ≤ d̂(x, y) + C_{1} + 2C_{2},

hence the image of the map

d̂(x, ·) : G ∩ p[x, y] → [0, ∞),

is contained in [0, d̂(x, y) + C_{1} + 2C_{2}]. Also, the image contains 0 and d̂(x, y). By Proposition 10(a), we have

|d̂(x, z′) − d̂(x, z)| ≤ M

when d(z′, z) = 1. This, together with the fact that p[x, y] is a geodesic path, implies that the image of the map

d̂(x, ·) : G ∩ p[x, y] → [0, d̂(x, y) + C_{1} + 2C_{2}]

is M-dense in [0, d̂(x, y)], i.e. for every t ∈ [0, d̂(x, y)], there exists a ∈ G ∩ p[x, y] such that

|d̂(x, a) − t| ≤ M.

It follows that d̂(x, a) ≤ t + M, and by Proposition 10(b) we also have

|d̂(x, y) − d̂(x, a) − d̂(a, y)| ≤ C_{1} + 2C_{2}.

This implies that

d̂(a, y) ≤ d̂(x, y) − d̂(x, a) + C_{1} + 2C_{2}
≤ d̂(x, y) − t + M + C_{1} + 2C_{2}.

Therefore (G, d̂) is weakly geodesic for δ_{1} := M + C_{1} + 2C_{2}.

Kasparov and Skandalis introduced the concept of bolicity in [11, 10].

Definition 15. A metric space (X, d) is said to be bolic if there exists δ_{2} ≥ 0 with the following properties: (B1) for any R > 0, there exists R′ > 0 such that for all a, a′, b, b′ ∈ X satisfying

d(a, a′) + d(b, b′) ≤ R  and  d(a, b) + d(a′, b′) ≥ R′,

we have

d(a, b′) + d(a′, b) ≤ d(a, b) + d(a′, b′) + 2δ_{2}; and

(B2) there exists a map m : X × X → X, such that, for all x, y, z ∈ X, we have

2d(m(x, y), z) ≤ (2d(x, z)^{2} + 2d(y, z)^{2} − d(x, y)^{2})^{\frac{1}{2}} + 4δ_{2}.

(X, d) is called strongly bolic if it is bolic and the above condition (B1) holds for every δ_{2} > 0 [13].

Proposition 16. The metric space (G, d̂) is strongly bolic.

Proof. Proposition 13 yields condition (B1) for all δ_{2} > 0. It remains to show that there exist δ_{2} ≥ 0 and a map m : G × G → G, such that, for all x, y, z ∈ G, we have

2d̂(m(x, y), z) ≤ (2d̂(x, z)^{2} + 2d̂(y, z)^{2} − d̂(x, y)^{2})^{\frac{1}{2}} + 4δ_{2}.

By Proposition 14 and its proof, there exists a vertex m(x, y) ∈ G ∩ p[x, y] such that

|d̂(x, m(x, y)) − \frac{d̂(x, y)}{2}| ≤ δ_{1} and |d̂(m(x, y), y) − \frac{d̂(x, y)}{2}| ≤ δ_{1}. (10)

By the definition of δ-hyperbolicity, we know that either

(1) there exists a ∈ G ∩ p[z, y] such that d(m(x, y), a) ≤ δ + 1, or

(2) there exists b ∈ G ∩ p[x, z] such that d(m(x, y), b) ≤ δ + 1.

In case (1), we have

|d̂(z, m(x, y)) − d̂(z, a)| ≤ d̂(m(x, y), a) ≤ δ + 1 + C_{1},
|d̂(y, m(x, y)) − d̂(y, a)| ≤ d̂(m(x, y), a) ≤ δ + 1 + C_{1}.

Hence, by Proposition 10(b), we obtain

d̂(z, m(x, y)) + d̂(x, y) ≤ d̂(z, a) + δ + 1 + C_{1} + d̂(x, y)

≤ d̂(z, a) + δ + 1 + C_{1} + d̂(x, m(x, y)) + d̂(m(x, y), y)

≤ d̂(z, a) + d̂(a, y) + d̂(x, m(x, y)) + 2δ + 2C_{1} + 2
≤ d̂(y, z) + d̂(x, m(x, y)) + δ′,

where δ′ := 2δ + 3C_{1} + 2. In case (2), we similarly have

d̂(z, m(x, y)) + d̂(x, y) ≤ d̂(x, z) + d̂(m(x, y), y) + δ′.

It follows from (10) that

d̂(z, m(x, y)) + d̂(x, y) ≤ sup {d̂(x, z) + d̂(y, m(x, y)), d̂(y, z) + d̂(x, m(x, y))} + δ′

≤ sup \{d̂(x, z), d̂(y, z)\} + \frac{d̂(x, y)}{2} + δ_{1} + δ′.

Hence

2d̂(z, m(x, y)) ≤ 2 sup {d̂(x, z), d̂(y, z)} − d̂(x, y) + 4δ_{2}, (11)

where δ_{2} := \frac{δ_{1}+δ′}{2} .

If t, u, and v are non-negative real numbers such that |u − v| ≤ t, then

(2u − v)^{2} ≤ 2u^{2} + 2t^{2} − v^{2}.

Setting t := inf{d̂(x, z), d̂(y, z)}, u := sup{d̂(x, z), d̂(y, z)}, v := d̂(x, y), we obtain

2 sup \{d̂(x, z), d̂(y, z)\} − d̂(x, y) ≤ (2d̂(x, z)^{2} + 2d̂(y, z)^{2} − d̂(x, y)^{2})^{\frac{1}{2}} .

Therefore, by (11),

2d̂(z, m(x, y)) ≤ (2d̂(x, z)^{2} + 2d̂(y, z)^{2} − d̂(x, y)^{2})^{\frac{1}{2}} + 4δ_{2}.

We summarize the results of this section.

Theorem 17. Every hyperbolic group G admits a metric d̂ with the following properties.

(1) d̂ is G-invariant, i.e. d̂(g · x, g · y) = d̂(x, y) for all x, y, g ∈ G.

(2) d̂ is quasiisometric to the word metric d, i.e. there exist A > 0 and B ≥ 0 such that

\frac{1}{A}d̂(x, y) − B ≤ d(x, y) ≤ Ad̂(x, y) + B

for all x, y ∈ G.

(3) The metric space (G, d̂) is weakly geodesic and strongly bolic.

6. The Baum-Connes conjecture for hyperbolic groups.

In this section, we combine Theorem 17 with Lafforgue’s work to prove the main result of this paper.

Definition 18. An action of a topological group G on a topological space X is called proper if the map G × X → X × X given by (g, x) ↦ (x, gx) is a proper map, that is the preimages of compact subsets are compact.

When G is discrete, an action is proper iff it is properly discontinuous, i.e. if the set {g ∈ G | K ∩ gK ≠ ∅} is finite for any compact K ⊆ X.

The following deep theorem was proved by Lafforgue using Banach KK-theory.

Theorem 19 (Lafforgue [13]). If a discrete group G has property RD, and G acts properly and isometrically on a strongly bolic, weakly geodesic, and uniformly locally finite metric space, then the Baum-Connes conjecture holds for G.

Theorem 20. The Baum-Connes conjecture holds for hyperbolic groups and their subgroups.

Proof. Let H be a subgroup of a hyperbolic group G. By Theorem 17(2), there exist constants A > 0 and B ≥ 0 such that d(a, b) ≤ A d̂(a, b) + B for all a, b ∈ G. Hence (G, d̂) is uniformly locally finite and the H-action on (G, d̂) is proper. By Theorem 17, (G, d̂) is weakly geodesic and strongly bolic, and the H-action on (G, d̂) is isometric. By a theorem of P. de la Harpe and P. Jolissaint, H has property RD [5, 9]. Now Theorem 19 implies Theorem 20.

Theorem 20 has been proved independently by Vincent Lafforgue using a different and elegant method [14].

The following result is a direct consequence of Theorem 20.

Theorem 21. The Kadison-Kaplansky conjecture holds for any torsion free subgroup G of a hyperbolic group, i.e. there exists no non-trivial projection in the reduced group C^{∗}- algebra C_{r}^{∗}(G).

Recall that an element p in C_{r}^{∗}(G) is said to be a projection if p^{∗} = p, p^{2} = p. A projection in C_{r}^{∗}(G) is said to be non-trivial if p ≠ 0, 1. It is well known that the Baum-Connes conjecture for a torsion free discrete group G implies the Kadison-Kaplansky conjecture for G [3, 2].

Michael Puschnigg has independently proved Theorem 21 using a beautiful local cyclic homology method [17]. Ronghui Ji has previously proved that there exists no non-trivial idempotent in the Banach algebra ℓ^{1}(G) for any torsion free hyperbolic group [8].

References

[1] J. M. Alonso, T. Brady, D. Cooper, T. Delzant, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short, Notes on word hyperbolic groups, in Group theory from a geometrical viewpoint, H. Short, ed., World Sci. Publishing, 1991, pp. 3–63.

[2] A. Baum, P. Connes and N. Higson, Classifying spaces for proper actions and K-theory for group C^{∗}-algebras, Volume 167, Contemporary Math. 241-291, Amer. Math. Soc., Providence, RI, 1994.

[3] P. Baum and A. Connes, K-theory for discrete groups, Vol. 1, volume 135 of London Math. Soc. Lecture Notes series, pages 1-20, Cambridge University Press, 1988.

[4] A. Connes and H. Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups. Topology 29 (1990), 345-388.

[5] P. de la Harpe, Groupes hyperboliques, algèbres d’opérateurs et un théorèm de Jolissaint. C.R.A.S., Paris, Série I(307): 771-774, 1988.

[6] M. Gromov, Hyperbolic groups, MSRI Publ. 8, 75-263, Springer, 1987.

[7] N. Higson and G. Kasparov, Operator K-theory for groups which act properly and isometrically on Hilbert space. Electronic Research Announcement, AMS 3 (1997), 131-141.

[8] R. Ji, Nilpotency of Connes’ periodicity operator and the idempotent conjectures. K-Theory 9 (1995), no. 1, 59-76.

[9] P. Jolissaint, Rapidly decreasing functions in reduced C^{∗}-algebras of groups. Trans. Amer. Math. Soc. 317: 167-196, 1990.

[10] G. Kasparov and G. Skandalis, Groupes boliques et conjecture de Novikov. C.R.A.S., Paris, Série I(319): 815-820, 1995.

[11] G. Kasparov and G. Skandalis, Groups acting properly on bolic spaces and the Novikov conjecture. Preprint 1998.

[12] V. Lafforgue, Compléments à la démonstration de la conjecture de Baum-Connes pour certains groupes possédant la propriété (T). C.R.A.S., Paris, Série I(328): 203-208, 1999.

[13] , K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes. To appear in Inventiones mathematicae, Ecole Normale Superior preprint 1998.

[14] , Private communication. 2001.

[15] , Une démonstration de la conjecture de Baum-Connes pour certains groupes possédant la propriété (T). C.R.A.S., Paris, Série I(327): 439-444, 1998.

[16] I. Mineyev, Straightening and bounded cohomology of hyperbolic groups. To appear in Geometric and Functional Analysis, Max-Planck-Institut preprint, also available at

http://www.math.usouthal.edu/~mineyev/math/.

[17] M. Puschnigg, The Kadison-Kaplansky conjecture for word hyperbolic groups. Preprint 2001.

University of South Alabama Dept of Mathematics and Statistics, ILB 325 Mobile, AL 36688, USA

mineyev@math.usouthal.edu
http://www.math.usouthal.edu/~mineyev/math/

Vanderbilt University Department of Mathematics 1326 Stevenson Center Nashville, TN 37240, USA

gyu@math.vanderbilt.edu