Journal of Lie Theory Volume 10 (2000) 255–267 ◯^{C}2000 Heldermann Verlag

A proof of property (RD) for cocompact lattices of SL(3,R) and SL(3,C)

Vincent Lafforgue

Communicated by A. Valette

Abstract. We prove that cocompact lattices in SL_{3}(ℝ) and SL_{3}(ℂ) have property (RD) of Jolissaint. This property asserts that functions on these groups which are l^{2} with some polynomial decay belong to the reduced C^{∗} -algebra. Ramagge, Robertson and Steger proved the same result for cocompact lattices in SL_{3} of p-adic fields and we use the same method.

In this article we prove that discrete cocompact subgroups of SL_{3}(ℝ) and SL_{3}(ℂ) satisfy property (RD) of Jolissaint (this property was introduced in [4, 5, 7]). The author originally proved the result for SL_{3}(ℝ) and the case of SL_{3}(ℂ) is due to the referee. The argument is a very close imitation of the argument of [9] : in this article Ramagge, Robertson and Steger prove a general result (stated below) implying property (RD) for finetely generated discrete subgroups of SL_{3}(𝔽), with 𝔽 a finite extension of ℚ_{p}. Our result is a special case of a conjecture of Valette which claims that any discrete group acting isometrically, properly and cocompactly either on a Riemannian symmetric space or on an affine building has property (RD) ([2] page 74).

Up to now property (RD) has been proved for free groups by Haagerup in [3], and then for hyperbolic groups by de la Harpe in [1], using [5]. Recently, in [9], Ramagge Robertson and Steger have proved property (RD) for any discrete group acting freely on the vertices of an Ã^{1} × Ã^{1} or Ã^{2} building by type-rotating automorphisms and this provided the first example of higher rank groups with property (RD). Our article is just an adaptation of [9] to SL_{3}(ℝ) and SL_{3}(ℂ) and it doesn’t bring any new idea. On the other hand a new idea is needed in order to prove property (RD) for cocompact lattices in Lie groups or p-adic groups of rank more than 1 and other than SL_{3}.

I thank Yves Benoist for many fruitful discussions.

1. General facts about property (RD)

Let Γ be a discrete group. A length l on Γ is a function l : Γ → ℝ_{+} such that l(1) = 0, l(g^{−1}) = l(g) and l(g_{1}g_{2}) ≤ l(g_{1}) + l(g_{2}). We write Γ_{r} = {g ∈ Γ, l(g) ≤ r}.

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag Definition 1.1. ([7]) We say that Γ satisfies (RD) with respect to a length l if there is a polynomial P such that, for any r ∈ ℝ_{+}, f_{1}, f_{2} ∈ ℂΓ with suppf_{1} ⊂ Γ_{r}, we have ∥f_{1} ∗ f_{2}∥_{l^{2}(Γ)} ≤ P (r)∥f_{1}∥_{l^{2}(Γ)}∥f_{2}∥_{l^{2}(Γ)}.

It is enough to check the inequality for f_{1}, f_{2} ∈ ℝ_{+}Γ.

Let us notice the following fact : if Γ satisfies (RD) with respect to a lenght l, there is a polynomial P such that, for any r ∈ ℝ_{+} and for any f ∈ ℂΓ with suppf ∈ Γ_{r}, one has ∥f∥_{l^{2}(Γ)} ≤ ∥f∥_{C_{r}^{∗}(Γ)} ≤ P (r)∥f∥_{l^{2}(Γ)}. This very good estimate for the norm in C_{r}^{∗}(Γ) has an important consequence : C_{r}^{∗}(Γ) has the same K-theory as a much simpler algebra, that we introduce now.

Proposition 1.2. If Γ satisfies (RD) w.r.t. l, if s ∈ ℝ_{+} is big enough, the completion H^{s}(Γ) of ℂΓ for the norm ∥f∥ = ( ∑_{g∈Γ} |f(g)|^{2}(1 + l(g))^{2s})^{1/2} is a Banach subalgebra of C_{r}^{∗}(Γ) which is dense and closed under holomorphic functional calculus. In [6] this is proved for the Jolissaint algebra H^{∞}(Γ) = ⋂ H^{s}(Γ). We give an adaptation of the proof to our case.

Proof. Let P be the polynomial in the definition. Take any s ∈ ℝ_{+} such that s > deg(P).

a) We prove that H^{s}(Γ) is a subspace of C_{r}^{∗}(Γ). We denote by χ_{0} the characteristic function of {g, l(g) ∈ [0, 1[} and for any n ∈ ℕ^{∗} we denote by χ_{n} the characteristic function of {g, l(g) ∈ [2^{n−1}, 2^{n}[}. For any f ∈ ℂΓ, ∥f∥_{C_{r}^{∗}(Γ)} ≤ ∑_{n=0}^{∞} ∥fχ_{n}∥_{C_{r}^{∗}(Γ)} ≤ ∑_{n=0}^{∞} P (2^{n})∥fχ_{n}∥_{l^{2}(Γ)} ≤ C∥f∥_{H^{s}(Γ)} with C = (P (1)^{2} +

∑_{n=1}^{∞}(P (2^{n})(1 + 2^{n−1})^{−s})^{2})^{\frac{1}{2}} by the Cauchy-Schwarz inequality.

b) We prove that H^{s}(Γ) is an algebra. For any f_{1}, f_{2} ∈ ℂΓ and for any g ∈ Γ we have

|(f_{1} ∗ f_{2})(g)|(1 + l(g))^{s} ≤ ∑_{\begin{matrix}g_{1},g_{2}, s.t. \\ g_{1}g_{2}=g \end{matrix}}2^{s}|f_{1}(g_{1})||f_{2}(g_{2})|((1 + l(g_{1}))^{s} + (1 + l(g_{2}))^{s})

and therefore

∥g ↦ |(f_{1} ∗ f_{2})(g)|(1 + l(g))^{s}∥_{l^{2}(Γ)}
≤ ∥g ↦ ( ∑_{\begin{matrix}g_{1},g_{2}, s.t. \\ g_{1}g_{2}=g \end{matrix}}2^{s}|f_{1}(g_{1})||f_{2}(g_{2})|(1 + l(g_{1}))^{s})∥_{l^{2}(Γ)}

+ ∥g ↦ ( ∑_{\begin{matrix}g_{1},g_{2}, s.t. \\ g_{1}g_{2}=g \end{matrix}}2^{s}|f_{1}(g_{1})||f_{2}(g_{2})|(1 + l(g_{2}))^{s})∥_{l^{2}(Γ)}.

The two terms are analoguous and for the first one we have

∥g ↦ ( ∑_{\begin{matrix}g_{1},g_{2}, s.t. \\ g_{1}g_{2}=g \end{matrix}}2^{s}|f_{1}(g_{1})||f_{2}(g_{2})|(1 + l(g_{1}))^{s})∥_{l^{2}(Γ)} ≤ 2^{s}C∥f_{1}∥_{H^{s}(Γ)}∥f_{2}∥_{H^{s}(Γ)}

by part a). c) Let t ∈]deg(P), s[. We first prove two intermediate results. α) H^{t}(Γ) is an algebra by part b) and H^{s}(Γ) is stable under holomorphic functional calculus in H^{t}(Γ). The proof is as follows. Since H^{s}(Γ) is dense in H^{t}(Γ), if x ∈ H^{s}(Γ) has an invertible image in H^{t}(Γ), there exists y ∈ H^{s}(Γ) such that the norms in H^{t}(Γ) of 1 − xy and 1 − yx are arbitrary small. Therefore we only have to prove that, for any f ∈ H^{s}(Γ),

lim_{n→∞} ∥f^{n}∥_{H^{s}(Γ)}^{1/n} = lim_{n→∞} ∥f^{n}∥_{H^{t}(Γ)}^{1/n}.

For any g ∈ Γ we have

|f^{n}(g)| ≤ ∑_{\begin{matrix}g_{1},...,g_{n}∈Γ, \\ g_{1},...,g_{n}=g \end{matrix}}|f(g_{1})| … |f(g_{n})|

and if g = g_{1}, …, g_{n}, (1 + l(g))^{s−t} ≤ n^{s−t}((1 + l(g_{1}))^{s−t} + ⋯ + (1 + l(g_{n}))^{s−t}). Therefore

∥f^{n}∥_{H^{s}(Γ)} = ∥g ↦ (1 + l(g))^{s−t}|f^{n}(g)|∥_{H^{t}(Γ)} ≤ n^{s−t+1}C′^{n−1}∥f∥_{H^{s}(Γ)}∥f∥_{H^{t}(Γ)}^{n−1},

where C′ is a constant such that ∥f_{1}f_{2}∥_{H^{t}(Γ)} ≤ C′∥f_{1}∥_{H^{t}(Γ)}∥f_{2}∥_{H^{t}(Γ)} for any f_{1}, f_{2} ∈ H^{t}(Γ). When n goes to infinity we get lim_{n→∞} ∥f^{n}∥_{H^{s}(Γ)}^{1/n} ≤ C′∥f∥_{H^{t}(Γ)} and the result easily follows by putting f^{p} instead of f in this inequality and making p go to infinity. β) For any f ∈ H^{s}(Γ) we have ∥f∥_{H^{t}(Γ)} ≤ ∥f∥H^{s}(Γ)^{\frac{t}{s}}∥f∥_{l^{2}(Γ)}^{1− \frac{t}{s}} by Hlder’s inequality.

Now let f ∈ H^{s}(Γ). We have to prove that f has the same spectral radius in H^{s}(Γ) and C_{r}^{∗}(Γ). If ρ_{H^{s}(Γ)}(f) = 0 this is obvious because ρ_{C_{r}^{∗}(Γ)}(f) ≤ ρ_{H^{s}(Γ)}(f).

Otherwise we have ∥f^{n}∥_{C_{r}^{∗}(Γ)} ≥ ∥f^{n}∥_{l^{2}(Γ)} ≥ ∥f^{n}∥H^{t}(Γ)^{\frac{s}{s−t}}∥f^{n}∥_{H^{s}(Γ)}^{− \frac{t}{s−t}} and the result

follows from α).

2. Analytical part of the proof

In this section we consider a discrete metric space (X, d) and a discrete group Γ acting freely and isometrically on X, and we introduce the groupoid G = X ×_{Γ} X such that G^{(0)} = Γ\X and G^{(1)} = Γ\X^{2} and we define G_{r} = {[x, y] ∈ G, d(x, y) ≤ r\} for any r ∈ ℝ_{+} and ∥f∥_{l^{2}(G)} = ( ∑_{g∈G} |f(g)|^{2})^{\frac{1}{2}} for any f ∈ ℂG.

We say that X and Γ satisfy the property P (X, Γ) if there is a polynomial P such that for any r ∈ ℝ_{+}, f_{1}, f_{2} ∈ ℝ_{+}G with suppf_{1} ∈ G_{r}, one has ∥f_{1} ∗ f_{2}∥_{l^{2}(G)} ≤ P (r)∥f_{1}∥_{l^{2}(G)}∥f_{2}∥_{l^{2}(G)}.

Proposition 2.1. If P (X, Γ) holds then Γ satisfies (RD) w.r.t. the lenght l(g) = d(x_{0}, gx_{0}) for any x_{0} ∈ X.

Proof. For any g_{1}, g_{2} ∈ Γ, [x_{0}, g_{1}x_{0}] ∘ [x_{0}, g_{2}x_{0}] = [x_{0}, g_{1}g_{2}x_{0}]. Let us define T : ℂΓ → ℂG by T (f)([x, y]) = 0 if x ∉ Γx_{0} or y ∉ Γx_{0} and T (f)([x_{0}, gx_{0}]) = f(g). For any f ∈ ℂΓ, ∥f∥_{l^{2}(Γ)} = ∥T (f)∥_{l^{2}(G)} and for f_{1}, f_{2} ∈ ℂΓ, T (f_{1} ∗_{Γ} f_{2}) = T (f_{1}) ∗_{G} T (f_{2}).

Definition 2.2. Let (Z, d) be a metric space and δ > 0. For any points x_{i} ∈ Z we say that x_{1} … x_{n} is a δ-path if d(x_{1}, x_{2}) + ⋯ + d(x_{n−1}, x_{n}) ≤ d(x_{1}, x_{n}) + δ and that x_{1}x_{2}x_{3} is a δ-thin triangle if there exists t ∈ Z such thatx_{1}tx_{2}, x_{2}tx_{3} and x_{3}tx_{1} are δ-paths. We say that (Z, d) satisfies (H_{δ}) if there exists a polynomial P such thatfor any r ∈ ℝ_{+}, x, y ∈ Z, one has

#{t ∈ Z, xty δ-path, d(x, t) ≤ r} ≤ P (r).

Proposition 2.3. Let δ > 0. If (X, d) satisfies (H_{δ}), there exists a polynomial P such that for any r ∈ ℝ_{+}, f_{1}, f_{2}, f_{3} ∈ ℝ_{+}G, with supp(f_{1}) ∈ G_{r}, one has

∑_{\begin{matrix}(x_{1},x_{2},x_{3})∈Γ\X^{3}, \\ x_{1}x_{2}x_{3} δ−thin \end{matrix}} f_{1}([x_{2}, x_{3}])f_{2}([x_{3}, x_{1}])f_{3}([x_{1}, x_{2}]) ≤ P (r)∥f_{1}∥_{l^{2}(G)}∥f_{2}∥_{l^{2}(G)}∥f_{3}∥_{l^{2}(G)}.

This proposition implies the result of [1] : hyperbolic groups satisfy property (RD). The following lemma is obvious.

Lemma 2.4. If H_{1}, H_{2}, H_{3} are Hilbert spaces, and T_{1} ∈ L(H_{3}, H_{2}), T_{2} ∈ L(H_{1}, H_{3}), T_{3} ∈ L(H_{2}, H_{1}) have finite Hilbert-Schmidt norms, |T r(T_{1}T_{2}T_{3})| ≤ ∥T_{1}∥_{HS}∥T_{2}∥_{HS}∥T_{3}∥_{HS}.

Proof of the proposition 2.3. We have

∑_{\begin{matrix}(x_{1},x_{2},x_{3})∈Γ\X^{3}, \\ x_{1}x_{2}x_{3} δ−thin \end{matrix}} f_{1}([x_{2}, x_{3}])f_{2}([x_{3}, x_{1}])f_{3}([x_{1}, x_{2}])

≤ ∑_{\begin{matrix}(x_{1},x_{2},x_{3},t)∈Γ\X^{4}, \\ x_{1}tx_{2},x_{2}tx_{3},x_{3}tx_{1} δ−paths \end{matrix}} f_{1}([x_{2}, x_{3}])f_{2}([x_{3}, x_{1}])f_{3}([x_{1}, x_{2}]).

Note that if x_{2}tx_{3} is a δ-path and d(x_{2}, x_{3}) ≤ r, then d(x_{2}, t) ≤ r + δ and d(x_{3}, t) ≤ r + δ.

Let H_{1}, H_{2}, H_{3} ⊂ l^{2}(Γ\X^{2}) be defined by H_{1} = l^{2}(Γ\X^{2}), H_{2} = l^{2}({(t, x_{2}) ∈ Γ\X^{2}, d(x_{2}, t) ≤ r + δ}) and H_{3} = l^{2}({(t, x_{3}) ∈ Γ\X^{2}, d(x_{3}, t) ≤ r + δ}), and let T_{1} ∈ L(H_{3}, H_{2}) be the operator defined as a matrix by

• T_{1,[t,x_{2}],[t′,x_{3}]} = f_{1}([x_{2}, x_{3}]) if t, t′ are in the same Γ-orbit (in this case we suppose t = t′) and if x_{2}tx_{3} is a δ-path

• and otherwise the coefficient is 0,

and let T_{2} and T_{3} be defined in the same way. We have

∑_{\begin{matrix}(x_{1},x_{2},x_{3},t)∈Γ\X^{4}, \\ x_{1}tx_{2},x_{2}tx_{3},x_{3}tx_{1} δ−paths \end{matrix}} f_{1}([x_{2}, x_{3}])f_{2}([x_{3}, x_{1}])f_{3}([x_{1}, x_{2}]) = T r(T_{1}T_{2}T_{3}),

but

∥T_{1}∥_{HS}^{2} ≤ ∑_{\begin{matrix}(x_{2},t,x_{3})∈Γ\X^{3}, \\ x_{2}tx_{3} δ−path, \\ d(x_{2},t)≤r+δ \end{matrix}} |f_{1}([x_{2}, x_{3}])|^{2} ≤ P (r + δ)∥f_{1}∥_{l^{2}(G)}^{2}

by (H_{δ}) and in the same way ∥T_{2}∥_{HS} ≤ \sqrt{P (r + δ)}∥f_{2}∥_{l^{2}(G)} and ∥T_{3}∥_{HS} ≤ \sqrt{P (r + δ)}∥f_{3}∥_{l^{2}(G)}. The proposition follows.

Let δ > 0. We say that X and Γ satisfy (K_{δ}) if there exist k ∈ ℕ and Γ-invariant subsets T_{1}, …, T_{k} of X^{3} such that

(K_{δ}a) there exists C_{1} ∈ ℝ_{+} such that for any (x_{1}, x_{2}, x_{3}) ∈ X^{3} , there exist i ∈ {1, …, k} and (t_{1}, t_{2}, t_{3}) ∈ T_{i} such that max (d(t_{1}, t_{2}), d(t_{2}, t_{3}), d(t_{3}, t_{1}))≤ C_{1}( min (d(x_{1}, x_{2}), d(x_{2}, x_{3}), d(x_{3}, x_{1})) + δ), and x_{1}t_{1}t_{2}x_{2}, x_{2}t_{2}t_{3}x_{3}, x_{3}t_{3}t_{1}x_{1} are δ-paths,

(K_{δ}b) for any i ∈ {1, …, k} and t_{1}, t_{2}, t_{3}, t′_{3} ∈ X, if (t_{1}, t_{2}, t_{3}) and (t_{1}, t_{2}, t′_{3}) are in T_{i} then the triangles t_{1}t_{3}t′_{3} and t_{2}t_{3}t′_{3} are δ-thin.

Theorem 2.5. If, for some δ > 0, X and Γ satisfy (H_{δ}) and (K_{δ}) then P (X, Γ) holds and therefore Γ satisfies property (RD).

Proof. Let G = X ×_{Γ} X, and f_{1}, f_{2}, f_{3} ∈ ℝ_{+}G with supp(f_{1}) ⊂ G_{r}.

We shall abbreviate (x_{1}, x_{2}, x_{3}) by x. For any i ∈ {1, ..., k}, let J_{i} denote the set of all t = (t_{1}, t_{2}, t_{3}) ∈ T_{i} satisfying max(d(t_{1}, t_{2}), d(t_{2}, t_{3}), d(t_{3}, t_{1})) ≤ C_{1}(r + δ), and for any t ∈ X^{3} let K(t) be the set of all (x_{1}, x_{2}, x_{3}) ∈ X^{3} for which d(x_{2}, t_{2}) ≤ r + δ, d(x_{3}, t_{3}) ≤ r + δ holds, and x_{1}t_{1}t_{2}x_{2}, x_{2}t_{2}t_{3}x_{3}, x_{3}t_{3}t_{1}x_{1} are δ-paths. For any x ∈ X^{3} such d(x_{2}, x_{3}) ≤ r, there exist i ∈ {1, ..., k} and t ∈ J_{i}, such that x ∈ K(t). Therefore we have

∑_{x∈Γ\X^{3}} f_{1}([x_{2}, x_{3}])f_{2}([x_{3}, x_{1}])f_{3}([x_{1}, x_{2}])

≤ ∑_{i=1}^{k} ∑_{t∈Γ\J_{i}} ∑_{x∈K(t)} f_{1}([x_{2}, x_{3}])f_{2}([x_{3}, x_{1}])f_{3}([x_{1}, x_{2}])

≤ ∑_{i=1}^{k} ∑_{t∈Γ\J_{i}} h_{1}([t_{2}, t_{3}])h_{2}([t_{3}, t_{1}])h_{3}([t_{1}, t_{2}]),

where h_{1} ∈ ℝ_{+}G is defined by

h_{1}([t_{2}, t_{3}]) = ( ∑_{\begin{matrix}(x_{2},x_{3})∈X^{2}, \\ x_{2}t_{2}t_{3}x_{3} δ−path, \\ d(x_{2},t_{2})≤r+δ \end{matrix}} f_{1}([x_{2}, x_{3}])^{2})^{1/2}

if d(t_{2}, t_{3}) ≤ C_{1}(r + δ) and h_{1}([t_{2}, t_{3}]) = 0 otherwise and h_{2}, h_{3} ∈ ℝ_{+}G are defined by similar expressions. The last inequality comes from lemma 2.4 with H_{1} = H_{2} = H_{3} = l^{2}(X) and T_{1} with coefficient T_{1,x_{2},x_{3}} = f_{1}([x_{2}, x_{3}]) if x_{2}t_{2}t_{3}x_{3} is a δ-path and d(x_{2}, t_{2}) ≤ r + δ and 0 otherwise and with T_{2} and T_{3} defined in the same way. But

∥h_{1}∥_{l^{2}(G)}^{2} ≤ ∑_{\begin{matrix}(x_{2},t_{2},t_{3},x_{3})∈Γ\X^{4}, \\ x_{2}t_{2}t_{3}x_{3} δ−path, \\ d(x_{2},t_{2})≤r+δ \\ d(t_{2},t_{3})≤C_{1}(r+δ) \end{matrix}} f_{1}([x_{2}, x_{3}])^{2} ≤ P (r + δ)P (C_{1}(r + δ))∥f_{1}∥_{l^{2}(G)}^{2}

and the same inequality holds for h_{2} and h_{3}.

Fix i ∈ {1, …, k} and replace C_{1}(r + δ) by r. It remains to show that there is a polynomial P such thatfor any r ∈ ℝ_{+} and h_{1}, h_{2}, h_{3} ∈ ℝ_{+}G, with support in G_{r}, we have

∑_{(t_{1},t_{2},t_{3})∈Γ\T_{i}} h_{1}([t_{2}, t_{3}])h_{2}([t_{3}, t_{1}])h_{3}([t_{1}, t_{2}]) ≤ P (r)∥h_{1}∥_{l^{2}(G)}∥h_{2}∥_{l^{2}(G)}∥h_{3}∥_{l^{2}(G)}.

But the sum is equal to ⟨h_{1} ∗_{T_{i}} h_{2}, h̃_{3}⟩_{l^{2}(G)} for some partial convolution along T_{i}, where h̃_{3}([x, y]) = \overline{h_{3}([y, x])}. We compute

⟨h_{1}∗_{T_{i}}h_{2}, h_{1}∗_{T_{i}}h_{2}⟩_{l^{2}(G)} = ∑_{\begin{matrix}(t_{1},t_{2},t_{3},t′_{3})∈Γ\X^{4}, \\ (t_{1},t_{2},t_{3})∈T_{i}, (t_{1},t_{2},t′_{3})∈T_{i} \end{matrix}} h_{1}([t_{2}, t_{3}])h_{2}([t_{3}, t_{1}]) \overline{h_{1}([t_{2}, t′_{3}])}\overline{h_{2}([t′_{3}, t_{1}])}.

By (K_{δ}b) the triangle t_{1}t_{3}t′_{3} and t_{2}t_{3}t′_{3} are δ-thin. By proposition 2.3 there is a polynomial P with

∥[t_{3}, t′_{3}] ↦ ∑_{\begin{matrix}t_{1}∈X, \\ t_{1}t_{3}t′_{3} δ−thin \end{matrix}} h_{2}([t_{3}, t_{1}])\overline{h_{2}([t′_{3}, t_{1}])}∥_{l^{2}(G)} ≤ P (r)∥h_{2}∥_{l^{2}(G)}^{2},

∥[t_{3}, t′_{3}] ↦ ∑_{\begin{matrix}t_{2}∈X, \\ t_{2}t_{3}t′_{3}, δ−thin \end{matrix}} h_{1}([t_{2}, t_{3}])\overline{h_{1}([t_{2}, t′_{3}])}∥_{l^{2}(G)} ≤ P (r)∥h_{1}∥_{l^{2}(G)}^{2},

and the theorem follows.

3. Geometrical part of the proof

3.1. The case of Ã^{2}-buildings. The following theorem is easily deducible from the arguments of [9].

Theorem 3.1. (extracted from [9]) If X is a free Γ-space and Z is the set of vertices of some Ã^{2}-building on which Γ acts by type rotating automorphisms, and d is the graph-theoretic distance on the 1-squeletton, and θ : X → Z is a surjective Γ-equivariant map such that sup_{z∈Z} #(θ^{−1}(z)) < +∞, and X is equipped with the distance θ^{∗}(d), then X and Γ satisfy (H_{0}) and (K_{0}) with k = 2, T_{1} ∪ T_{2} the set of (t_{1}, t_{2}, t_{3}) ∈ X^{3} such that θ(t_{1})θ(t_{2})θ(t_{3}) is an equilateral triangle in some apartment, and (t_{1}, t_{2}, t_{3}) ∈ T_{1} if (θ(t_{1}), θ(t_{2})) is of shape (p, 0), p ∈ ℕ, and (t_{1}, t_{2}, t_{3}) ∈ T_{2} if (θ(t_{1}), θ(t_{2})) is of shape (0, p), p ∈ ℕ^{∗}. Consequently Γ satisfies property (RD).

We thus obtain a very slight improvement of the result of [9].

Corollary 3.2. Any discrete group Γ acting on the set X of vertices of an Ã^{2}-building by type-rotating automorphisms satisfies property (RD), provided that sup_{x∈X} #{g ∈ Γ, gx = x} < +∞.

3.2. The case of SL_{3}(R).

It is helpful to write 3 × 3-diagonal matrices with diagonal entries a, b, and c as D(a, b, c). We now consider G = SL_{3}(ℝ), K = SO_{3}(ℝ),

A = {D(e^{α_{1}}, e^{α_{2}}, e^{α_{3}}), α_{1} + α_{2} + α_{3} = 0} and
\overline{A^{+}} = \{D(e^{α_{1}}, e^{α_{2}}, e^{α_{3}}), α_{1} ≥ α_{2} ≥ α_{3}, α_{1} + α_{2} + α_{3} = 0\}.

We equip G/K with the distance d(x, y) = log ∥x^{−1}y∥ + log ∥y^{−1}x∥. We remark that d(x, y) = ρ log a if x^{−1}y ∈ KaK with a in \overline{A^{+}}, and ρ is defined by ρ(α_{1}, α_{2}, α_{3}) = α_{1} − α_{3}.

For any t ∈ ℝ, x, y, z ∈ G/K, we say that (x, y) is of shape (t, 0) if x^{−1}y ∈ Ke^{− \frac{t}{3}} D(e^{t}, 1, 1)K and that (x, y, z) is an equilateral triangle of oriented size t if there exists g ∈ G such that

x = gK, y = ge^{− \frac{t}{3}} D(e^{t}, 1, 1)K and z = ge^{− \frac{2t}{3}} D(e^{t}, e^{t}, 1)K.

If t ∈ ℝ_{−} and (x, y) is of shape (t, 0) we say also that (x, y) is of shape (0, −t) : in this way our terminology completely agrees with [9]. For any δ_{0} > 0 and for j = 1, 2 we let T_{j,δ_{0}} denote the set of all (t_{1}, t_{2}, t_{3}) ∈ (G/K)^{3} for which there exists a t in ℝ_{+} if j = 1, and in ℝ_{−} if j = 2, and for which there is a triple (s_{1}, s_{2}, s_{3}) ∈ (G/K)^{3} such that s_{1}s_{2}s_{3} is an equilateral triangle of oriented size t, and that d(s_{1}, t_{1}) ≤ δ_{0}, d(s_{2}, t_{2}) ≤ δ_{0}, and d(s_{3}, t_{3}) ≤ δ_{0}.

Theorem 3.3. Let Γ be a discrete subgroup of G = SL_{3}(ℝ), Z a Γ-invariant discrete subspace of G/K, and r ∈ ℝ_{+} such that the two following conditions are fulfilled :

(I1) ⋃_{x∈Z} B(x, r) = G/K

(I2) for any R ∈ ℝ_{+} sup_{x∈G/K} #(B(x, R) ∩ Z) is finite.

Let X be a free Γ-space and θ : X → Z a Γ-equivariant map such that sup_{z∈Z} #(θ^{−1}(z)) < +∞, and equip X with the distance θ^{∗}(d).

Then X and Γ satisfy (H_{δ}) and (K_{δ}) for some δ > 0 and with k = 2 and T_{1} = θ^{−1}(T_{1,r}) = {(t_{1}, t_{2}, t_{3}) ∈ X^{3}, (θ(t_{1}), θ(t_{2}), θ(t_{3})) ∈ T_{1,r}} and T_{2} = θ^{−1}(T_{2,r}). Consequently Γ satisfies property (RD).

If Γ is a discrete cocompact subgroup of SL_{3}(ℝ), every Γ-orbit Z in G/K satisfies (I1) and (I2), and we can choose X = Γ and θ obvious, and therefore Γ satisfies (RD).

Proof of the theorem. This proof is everything until part 4.

a) We first prove that X satisfies property (H_{δ}) for any δ > 0. This part of the argument works for any linear connected semi-simple Lie group.

We recall some notations from chapter 5 of [8]. Let G be a linear connected semi-simple Lie group, K a maximal compact subgroup, 𝔤 = 𝔨 ⊕ 𝔭 the decomposition associated to the Cartan involution, 𝔞 a maximal abelian subspace of 𝔭, Σ the set of restricted roots, 𝔤_{λ} the root space associated to λ ∈ Σ, Σ^{+} a choice of a set of positive roots, 𝔞^{+} = {H ∈ 𝔞, , λ(H) > 0 for all λ ∈ Σ^{+}}, A = exp(𝔞), A^{+} = exp(𝔞^{+}), and ρ = \frac{1}{2} ∑_{λ∈Σ^{+}}(dim𝔤_{λ})λ. We have G = K\overline{A^{+}}K and

∫_{G} f(g)dg = ∫_{K×A^{+}×K} f(kak′) ∏_{λ∈Σ^{+}}(sinh λ(log a))^{dim𝔤_{λ}}dkdadk′  (1)

is the integral formula corresponding to this decomposition. If ω_{1}, …, ω_{k} are the fundamental weights, we have ρ = n_{1}ω_{1} + ⋯ + n_{k}ω_{k} for some positive integers n_{1}, …, n_{k} depending on the multiplicities of the roots. We introduce the nonsymmetric fonction d_{i} on (G/K)^{2} : if x, y ∈ G/K and x^{−1}y = KaK with a ∈ \overline{A^{+}}, d_{i}(x, y) = ω_{i}(log a). Since G admits a representation of highest weight a multiple of ω_{i}, if we choose an hermitian metric on this representation compatible with the Cartan involution on G, for any x, y ∈ G/K, d_{i}(x, y) is a fraction of the log of the norm of the image by this representation of any antecedent of x^{−1}y in G. Therefore, for any x, y, z ∈ G/K, d_{i}(x, z) ≤ d_{i}(x, y) + d_{i}(y, z). Let α_{1}, …, α_{k} ∈ ℝ_{+}^{∗}, and consider the non-symmetric function d_{α}(x, y) = ∑_{i=1}^{k} α_{i}d_{i}(x, y). Up to a constant there is a unique G-invariant element of volume on G/K.

Lemma 3.4. For any δ > 0 there is a polynomial P such that for any r ∈ ℝ_{+} and x, y ∈ G/K,

Vol{t ∈ G/K, d_{α}(x, t) + d_{α}(t, y) ≤ d_{α}(x, y) + δ and d_{α}(x, t) ≤ r} ≤ P (r).

The lemma is false if some α_{i} is 0 : in this case the best estimate for the volume grows exponentially in r.

Proof. We denote by d the following distance on G/K : d(x, y) = ∑_{i=1}^{k} n_{i}d_{i}(x, y) = ρ log(a) if x^{−1}y = KaK with a ∈ \overline{A^{+}}.

Denote by 1 the origin in G/K. We may assume x = 1. There exists some constant C_{0} depending on α such that the conditions d_{α}(1, t) + d_{α}(t, y) ≤ d_{α}(1, y) + δ and d_{α}(1, t) ≤ r imply d_{i}(1, t) + d_{i}(t, y) ≤ d_{i}(1, y) + C_{0}δ for any i and d(1, t) ≤ C_{0}r. Because of (1) there exists some constant C_{1} ∈ ℝ_{+}^{∗} such that

Vol{z ∈ G/K, ∃k ∈ K, d(y, kz) ≤ 1} ≥ C_{1}e^{2d(1,y)}.

Therefore

Vol{(t, z) ∈ (G/K)^{2}, ∀i, d_{i}(1, t) + d_{i}(t, z) ≤ d_{i}(1, y) + C_{0}δ + 1, d(1, t) ≤ C_{0}r}

≥ C_{1}e^{2d(1,y)}Vol{t ∈ G/K, ∀i, d_{i}(1, t) + d_{i}(t, y) ≤ d_{i}(1, y) + C_{0}δ, d(1, t) ≤ C_{0}r}

because {t ∈ (G/K), ∀i, d_{i}(1, t) + d_{i}(t, z) ≤ d_{i}(1, y) + C_{0}δ + 1, d(1, t) ≤ C_{0}r} ⊃

{t ∈ (G/K), ∀i, d_{i}(1, t) + d_{i}(t, y) ≤ d_{i}(1, y) + C_{0}δ, d(1, t) ≤ C_{0}r} if d(y, z) ≤ 1 and

Vol{t ∈ (G/K), ∀i, d_{i}(1, t) + d_{i}(t, z) ≤ d_{i}(1, y) + C_{0}δ + 1, d(1, t) ≤ C_{0}r} depends only on Kz in K\G/K. The following fact comes from (1) : there exists a constant C_{2} such that

for any a_{1}, …, a_{k} ∈ ℝ_{+}, Vol{u ∈ G/K, ∀i, d_{i}(1, u) ≤ a_{i}} ≤ C_{2}e^{2 ∑ n_{i}a_{i}}.

Now fix t ∈ G/K such that d_{i}(1, t) ≤ C_{0}r. We have

Vol{z ∈ (G/K), ∀i, d_{i}(1, t) + d_{i}(t, z) ≤ d_{i}(1, y) + C_{0}δ + 1}
= Vol{z ∈ (G/K), ∀i, d_{i}(t, z) ≤ d_{i}(1, y) + C_{0}δ + 1 − d_{i}(1, t)}
≤ C_{2} exp (2d(1, y) + 2(C_{0}δ + 1)(∑_{i=1}^{k} n_{i}) − 2d(1, t)),

and ∫_{t∈G/K,d(1,t)≤C_{0}r} e^{−2d(1,t)}dt ≤ P (r) for some polynomial P, because of (1). The lemma now results from Fubini’s lemma.

We now come back to the notations of the theorem 3.3. We fix δ > 0 and we explain why X satisfies (H_{δ}). Of course it is enough to prove that (Z, d) satisfies (H_{δ}). By (I2) there exists N ∈ ℕ such that #(B(x, 1) ∩ Z) ≤ N for any x ∈ G/K. For any r ∈ ℝ_{+}, x, y ∈ G/K,

#{t ∈ Z, xty δ-path, d(x, t) ≤ r} ≤
\frac{N}{V} Vol\{t′ ∈ G/K, xt′y (δ + 2)-path, d(x, t′) ≤ r + 1\},

where V = Vol{z ∈ G/K, d(1, z) ≤ 1}, because if d(t, t′) ≤ 1, d(x, t) ≤ r and xty is a δ-path, then xt′y is a (δ + 2)-path and d(x, t′) ≤ r + 1. It remains to apply lemma 3.4 with d_{α} = d.

b) We now prove that X and Γ satisfy (K_{δ}) with k = 2, T_{1} and T_{2} as in the theorem 3.3 and if δ is big enough. This part of the proof really use fact that the flats in G/K are of type A^{2}.

The following lemma is analoguous to the study of the foldings in [9]. We call a flat in G/K any subset of G/K equal to gAK, for some g ∈ G. We now study the distance of some fixed point of G/K to the points of a fixed flat in G/K. Up to left translation by an element of G, we may suppose that the flat is AK. We prove the following result : there is some δ > 0 such that for any x ∈ G/K there exist y, y_{2}, y_{3} in G/K such that, for any a ∈ A, xy(aK) is a δ-path, and such that yy_{2}y_{3} is an equilateral triangle, y_{2} and y_{3} are at distance less than δ from AK and for any a ∈ A there exists some point z on the side y_{2}y_{3} of the triangle yy_{2}y_{3} such that yz(aK) is a δ-path.

Denote by W_{0} the subgroup of G whose elements are

Id, (\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}) , and (\begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) .

In the following lemma we shall again use the abbreviation for diagonal matrices:

D(a, b, c) = (\begin{matrix}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix}) .

Lemma 3.5. For some δ > 0 the following is true. For any x ∈ G/K there exist y ∈ G/K, t ∈ ℝ_{+}, m ∈ W_{0}A such that

(i) xy(aK) is a δ-path for any a ∈ A,

(ii) |d(y, mD(e^{s_{1}}, e^{s_{2}}, e^{s_{3}})K)−t−max(|s_{2} −s_{3}|, s_{1} −s_{2} −t, s_{1} −s_{3} −t, s_{2} −s_{1}, s_{3} − s_{1})| ≤ δ for any s_{1}, s_{2}, s_{3} ∈ ℝ, s_{1} + s_{2} + s_{3} = 0,

(iii) there exists h_{2} ∈ G such that d(h_{2}D(e^{s_{1}}, e^{s_{2}}, e^{s_{3}})K, mD(e^{s_{1}}, e^{s_{2}}, e^{s_{3}})K) ≤ δ if s_{1} + s_{2} + s_{3} = 0 and s_{2} ≥ s_{3} and d(h_{2}e^{− \frac{2t}{3}} D(e^{t}, 1, e^{t})K, y) ≤ δ,

(iv) there exists h_{3} ∈ G such that d(h_{3}D(e^{s_{1}}, e^{s_{2}}, e^{s_{3}})K, mD(e^{s_{1}}, e^{s_{2}}, e^{s_{3}})K) ≤ δ if s_{1} + s_{2} + s_{3} = 0 and s_{2} ≤ s_{3} and d(h_{3}e^{− \frac{2t}{3}} D(e^{t}, e^{t}, 1)K, y) ≤ δ.

We notice that the second condition is in fact a consequence of the two last ones (for a different δ).

264 Lafforgue Proof. Up to left translation by some element m ∈ W_{0}A we may suppose that

x = λ (\begin{matrix}e_{1} \\ e_{2} \\ e_{3} \end{matrix}) K, with λ ∈ ℝ_{+}^{∗}, e_{1}, e_{2}, e_{3} ∈ ℝ^{3}, ∥e_{1}∥ = ∥e_{2}∥ = ∥e_{3}∥ = 1 and

∥e_{2} ∧ e_{3}∥ ≤ min(∥e_{1} ∧ e_{2}∥, ∥e_{1} ∧ e_{3}∥). We have

|∥e_{1} ∧ e_{2}∥ − ∥e_{1} ∧ e_{3}∥| ≤ min_{μ=±1} ∥e_{1} ∧ (e_{3} − μe_{2})∥

≤ min_{μ=±1} ∥e_{3} − μe_{2}∥ ≤ \sqrt{2}∥e_{2} ∧ e_{3}∥.

Denote by ν an element of {−1, 1} where the minimum of ∥e_{3} − μe_{2}∥ is reached. Take t = log(\frac{∥e_{1}∧e_{2}∥}{∥e_{2}∧e_{3}∥}) ∈ ℝ_{+}. By the last inequalities |t−log(\frac{∥e_{1}∧e_{3}∥}{∥e_{2}∧e_{3}∥})| ≤ log(1+\sqrt{2}). For a = D(e^{s_{1}}, e^{s_{2}}, e^{s_{3}}), with s_{1} + s_{2} + s_{3} = 0, we have

d(aK, x) = log ∥a^{−1} (\begin{matrix}e_{1} \\ e_{2} \\ e_{3} \end{matrix}) ∥ + log ∥ (\begin{matrix}e_{1} \\ e_{2} \\ e_{3} \end{matrix})^{−1} a∥

and

| log ∥a^{−1} (\begin{matrix}e_{1} \\ e_{2} \\ e_{3} \end{matrix}) ∥ − max(−s_{1}, −s_{2}, −s_{3})| ≤ log 3

and since

(\begin{matrix}e_{1} \\ e_{2} \\ e_{3} \end{matrix})^{−1} = det(e_{1}, e_{2}, e_{3})^{−1} (\begin{matrix}e_{2} ∧ e_{3} \\ e_{3} ∧ e_{1} \\ e_{1} ∧ e_{2} \end{matrix})^{t}

we have

| log ∥ (\begin{matrix}e_{1} \\ e_{2} \\ e_{3} \end{matrix})^{−1} a∥ + log ∥e_{1} ∧ e_{2} ∧ e_{3}∥

− max(s_{1} + log ∥e_{2} ∧ e_{3}∥, s_{2} + log ∥e_{1} ∧ e_{3}∥, s_{3} + log ∥e_{1} ∧ e_{2}∥)| ≤ log 3. Therefore

|d(aK, x) − max(−s_{1}, −s_{2}, −s_{3}) − max(s_{1} − t, s_{2}, s_{3})
+ log ∥e_{1} ∧ e_{2} ∧ e_{3}∥ − log ∥e_{1} ∧ e_{2}∥| ≤ δ_{0}

for some numerical constant δ_{0}.

Now consider

y = e^{\frac{t}{3}} (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & ν & e^{−t} \end{matrix}) K.

By applying the last argument to y instead of x, we obtain |d(aK, y) − max(−s_{1}, −s_{2}, −s_{3}) − max(s_{1} − t, s_{2}, s_{3}) − t| ≤ δ_{1} for some numerical constant δ_{1} (different from δ_{0} because one has to normalize (0, ν, e^{−t})). But

d(x, y) = log ∥(\begin{matrix}e_{1} \\ e_{2} \\ e^{t}(e_{3} − νe_{2}) \end{matrix})∥ + log ∥(\begin{matrix}e_{1} \\ e_{2} \\ e^{t}(e_{3} − νe_{2}) \end{matrix})^{−1}∥

since

(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & ν & e^{−t} \end{matrix})^{−1} = (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & −νe^{t} & e^{t} \end{matrix}) .

Therefore

|d(x, y) − log ∥e_{1} ∧ e_{2}∥ + t + log ∥e_{1} ∧ e_{2} ∧ e_{3}∥| ≤ δ_{2}

for some numerical constant δ_{2} because ∥e^{t}(e_{3} − νe_{2})∥ ≤ \sqrt{2}∥e_{1} ∧ e_{2}∥ and ∥e_{2} ∧ (e^{t}(e_{3} − e_{2}))∥ = ∥e_{1} ∧ e_{2}∥ and ∥e_{1} ∧ e^{t}(e_{3} − νe_{2})∥ ≤ ∥e^{t}(e_{3} − νe_{2})∥.

Thus the first two assertions are proved. Now we take

h_{2} = (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & ν^{−1} \\ 0 & 0 & 1 \end{matrix}) and h_{3} = (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & ν & 1 \end{matrix}) .

We check that

h_{3}e^{− \frac{2t}{3}} D(e^{t}, e^{t}, 1)K = y and that d(h_{2}e^{− \frac{2t}{3}} D(e^{t}, 1, e^{t})K, y)

is bounded by a numerical constant, because

y^{−1}h_{2}e^{− \frac{2t}{3}} (\begin{matrix}e^{t} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & e^{t} \end{matrix}) K = K (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & −νe^{t} & e^{t} \end{matrix}) (\begin{matrix}1 & 0 & 0 \\ 0 & e^{−t} & ν^{−1} \\ 0 & 0 & 1 \end{matrix}) K

= K (\begin{matrix}1 & 0 & 0 \\ 0 & e^{−t} & ν^{−1} \\ 0 & −ν & 0 \end{matrix}) K.

Finally it is obvious that

d(h_{2}D(e^{s_{1}}, e^{s_{2}}, e^{s_{3}})K, D(e^{s_{1}}, e^{s_{2}}, e^{s_{3}})K) ≤ δ

if s_{1} + s_{2} + s_{3} = 0 and s_{2} ≥ s_{3}, and if δ is big enough. The last condition for h_{3} is similar.

Now consider x_{1}, x_{2}, x_{3} ∈ G/K. Take x = x_{1} and choose any flat containing x_{2} and x_{3}. After a small discussion of the possible positions of x_{2} and x_{3} in this flat (a similar discussion occurs in [9]) we obtain the following lemma which immediately implies that property (K_{δ}) holds with k = 2 and T_{1} and T_{2} as in the theorem 3.3 if δ is big enough.

Lemma 3.6. For some δ > 0 the following is true. For any x_{1}, x_{2}, x_{3} ∈ G/K there exist t_{1}, t_{2}, t_{3} ∈ G/K such that x_{1}t_{1}t_{2}x_{2}, x_{2}t_{2}t_{3}x_{3} and x_{3}t_{3}t_{1}x_{1} are δ-paths and t_{1}t_{2}t_{3} is an equilateral triangle.

c) Now we prove that property (K_{δ}b) holds with k = 2 and T_{1} and T_{2} as in the theorem if δ is big enough.

This results from the following lemma, applied to δ_{0} = 2r.

Lemma 3.7. For any δ_{0} > 0 there exists δ > 0 such that the following is true. For any s, t ∈ ℝ of the same sign, and x_{1}, y_{1}, x_{2}, z_{2} ∈ G/K with d(x_{1}, x_{2}) ≤ δ_{0}, (x_{1}, y_{1}) of shape (s, 0) and (x_{2}, z_{2}) of shape (t, 0), then x_{1}y_{1}z_{2} is a δ-thin triangle.

Proof. By symmetry with respect to 1 in G/K we may suppose s, t ∈ ℝ_{+}. Up to left translation by an element of G we may suppose x_{1} = 1 and y_{1} = e^{− \frac{s}{3}} D(e^{s}, 1, 1)K. We have z_{2} = he^{− \frac{t}{3}} D(e^{t}, 1, 1)K with h ∈ G such that d(1, hK) ≤ δ_{0}. Write

h = (\begin{matrix}h_{11} & h_{12} & h_{13} \\ h_{21} & h_{22} & h_{23} \\ h_{31} & h_{32} & h_{33} \end{matrix}) and h^{−1} = (\begin{matrix}h′_{11} & h′_{12} & h′_{13} \\ h′_{21} & h′_{22} & h′_{23} \\ h′_{31} & h′_{32} & h′_{33} \end{matrix}) .

By Cramer’s formula there exists a constant δ_{1} depending only on δ_{0} such that | log(max(|h_{21}|, |h_{31}|)) − log(max(|h′_{21}|, |h′_{31}|))| ≤ δ_{1}. Take r = min(s, t, − log(max(|h_{21}|, |h_{31}|)) and u = e^{− \frac{r}{3}} D(e^{r}, 1, 1)K. Then x_{1}uy_{1}, y_{1}uz_{2} and z_{2}ux_{1} are δ-paths for some δ depending only on δ_{0}. Indeed d(x_{1}, u) = r, d(u, y_{1}) = s − r, and since

u^{−1}z_{2} = Ke^{\frac{r−t}{3}} D(e^{−r}, 1, 1)hD(e^{t}, 1, 1)K

= Ke^{\frac{r−t}{3}} (\begin{matrix}e^{t−r}h_{11} & e^{−r}h_{12} & e^{−r}h_{13} \\ e^{t}h_{21} & h_{22} & h_{23} \\ e^{t}h_{31} & h_{32} & h_{33} \end{matrix}) K

and z_{2}^{−1}u = Ke^{\frac{t−r}{3}} D(e^{−t}, 1, 1)h^{−1}D(e^{r}, 1, 1)K

= Ke^{\frac{t−r}{3}} (\begin{matrix}e^{r−t}h′_{11} & e^{−t}h′_{12} & e^{−t}h′_{13} \\ e^{r}h′_{21} & h′_{22} & h′_{23} \\ e^{r}h′_{31} & h′_{32} & h′_{33} \end{matrix}) K,

we see that |d(u, z_{2}) − (t − r)| ≤ δ_{2} and |d(y_{1}, z_{2}) − (t − r) − (s − r)| ≤ δ_{2} by the same computation, where δ_{2} depends only on δ_{0}.

The lemmas 3.5 and 3.7 are more intuitive if one considers quadratic forms on ℝ^{3} instead of elements of SL_{3}(ℝ) but it is more difficult to write correct proofs in this way.

4. The case of SL_{3}(ℂ)

Very few things need to be changed. We put G = SL_{3}(ℂ) and K = SU_{3}(ℂ) instead of SL_{3}(ℝ) and SO_{3}(ℝ) but A is the same. The part a) of the proof of theorem 3.3 is true for any linear connected semi-simple group. In the part b), W_{0} remains the same and the lemma 3.5 is still true. In the proof of lemma 3.5 we have to take μ and ν in {z ∈ ℂ, |z| = 1} instead of {−1, 1}. Lemma 3.7 and its proof remain the same.

It would be interesting to know wether the lemmas 3.5 and 3.7 are still true for the groups SL(3, ℍ) and E_{6(−26)}, whose associated symetric spaces have also flats of type A^{2}.

References

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Département de Mathématiques et Applications Ecole Normale Supérieure 45 rue d’Ulm 75230 PARIS CEDEX 05, France vlafforg@dmi.ens.fr

Received September 25, 1998 and in final form May 1, 1999