PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 7, July 1998, Pages 1941–1944 S 0002-9939(98)04453-0

A NEW PROOF OF THE SOLOMON-TITS THEOREM

CHAO KU

(Communicated by Ronald M. Solomon)

Abstract. We give a new proof of the Solomon-Tits Theorem which asserts that the Tits building of a finite group of Lie type has the homotopy type of a bouquet of spheres.

1. Introduction

Let G be a finite group of Lie type of rank n ⩾ 3, (G, B, N, S) a Tits system for G, H = B ∩ N, B = UH where U = O_{p}(B) and p is the defining prime for G. F = {G_{i}; 1 ⩽ i ⩽ n} is the set of maximal parabolics of G over B and B = C(G, F) the simplicial complex whose vertex set is the union of coset spaces G/G_{i}, i ∈ I, and whose simplices are collection of vertices with nonempty intersection. B is called the Tits building of G. The following theorem is well known:

Theorem (Solomon-Tits). The Tits building B has the homotopy type of a wedge of |U| (n − 1)-spheres.

In his seminar paper [S] Solomon first mentioned the theorem and sketched a proof. Later Garland ([G]) gave a proof which is essentially the same as Solomon’s, and Curtis and Lehrer ([CL]) gave a proof of a related result based on the computation of the ℚG-endomorphism algebra of H_{∗}(B). We provide a new proof here which should help to better understand the structure of the Tits building. Namely, B is the union of the |U| apartments containing a fixed chamber, each having the homotopy type of an (n − 1)-sphere as it is a Coxeter complex. We show that the intersection of any k (k ⩾ 2) of these subcomplexes is contractible. Then it follows from an easy lemma in topology that B has the homotopy type of a wedge of these spheres.

2. Preliminary lemmas

Let us begin by recording a well known lemma from topology.

Lemma 1. Suppose X is a simplicial complex and X_{1}, X_{2}, …, X_{m}, m > 1, are subcomplexes of X with X = ⋃_{1⩽i⩽m} X_{i} such that for any 1 ⩽ i_{1} < i_{2} < ⋯ < i_{k} ⩽ m, k > 1, the set ⋂_{1⩽j⩽k} X_{i_{j}} is contractible. Then X has the homotopy type of X_{1} ∨ X_{2} ∨ ⋯ ∨ X_{m}.

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors December 19, 1996.

1991 Mathematics Subject Classification. Primary 20E42.

Proof: By induction. Note that if B is a contractible subcomplex of a simplicial complex A, then the quotient map q : A → A/B is a homotopy equivalence. If m = 2, as Y = X_{1} ∩X_{2} is contractible, X/Y = X_{1}/Y ∪X_{2}/Y and X_{1}/Y ∩X_{2}/Y = Y/Y is a point, the lemma follows easily in this case.

Assume the lemma holds for < m. Then X_{1} ∩ (⋃_{i>1} X_{i}) = ⋃_{i>1}(X_{1} ∩ X_{i}) is homotopy equivalent to ⋁_{i>1}(X_{1} ∩ X_{i}) by induction and hence contractible. Therefore X is homotopy equivalent to X_{1} ∨ (⋃_{i>1} X_{i}), which is in turn homotopy equivalent to X_{1} ∨ X_{2} ∨ ⋯ ∨ X_{m}.

We refer readers who are unfamiliar with the geometry of groups of Lie type and the theory of Coxeter groups to [A] or [B]. From time to time by abuse of notation we identify an element of N with its image in the Weyl group W = N/H. We may also use the same notation for a simplex of a complex and the subcomplex consisting of its faces.

Lemma 2. Let W_{J} = G_{J} ∩ N/H be a parabolic subgroup of W, where G_{J} = ⋂_{j∈J⊆I} G_{i}. Then each coset W_{J}d has a unique element d_{J} of minimal length such that l(wd_{J}) = l(w) + l(d_{J}) for all w ∈ W_{J}. Here l is the length function on W with respect to S.

Proof. This is exercise 3, page 37 in [B].

Define a partial ordering ⩽ on W by v ⩽ w if and only if w = uv and l(w) = l(u) + l(v) for some u ∈ W. This is a well defined ordering, and w_{0}, the longest element of W, is the unique maximal element with respect to this order. See for instance exercise 6, page 155 in [A].

Lemma 3. Let Φ be the Coxeter complex (see page 210 in [A] for the definition) for W, C ∈ Φ the fundamental chamber and σ ∈ Φ a simplex. Then there is a unique w_{σ} ∈ W with σ ⊆ Cw_{σ} such that for any chamber Cw containing σ, w_{σ} ⩽ w. Moreover, there is a sequence of chambers Cw_{σ} = Cw_{1}, Cw_{2}, …, Cw_{n} = Cw, each containing σ, such that w_{i} ⩽ w_{i+1} with l(w_{i+1}) = l(w_{i}) + 1, 1 ⩽ i < n.

Proof. This is a direct consequence of Lemma 2.

3. Proof of the Theorem

Let C = {G_{i}; 1 ⩽ i ⩽ n} be the fundamental chamber of B and Σ = ⋃_{w∈W} Cw. Then Σ is an apartment stabilized by N isomorphic to the Coxeter complex Φ. Hence, as Φ has the homotopy type of a (n − 1)-sphere,

Proposition 1. Σ has the homotopy type of a (n − 1)-sphere.

As G = BNB = BNU, it is obvious that

Proposition 2. B = ⋃_{u∈U} Σ^{u}, where Σ^{u} = ⋃_{w∈W} Cwu.

Notice that for u_{i} ∈ U, 1 ⩽ i ⩽ k, ⋂_{1⩽i⩽k} Σ^{u_{i}} is isomorphic to

( ⋂_{1⩽i⩽k} Σ^{u_{i}})^{u_{1}^{−1}} = ⋂_{1⩽i⩽k} Σ^{u_{i}u_{1}^{−1}}.

So by the argument in the introduction, a proof of the following proposition will complete the proof of the Solomon-Tits Theorem.

Proposition 3. For distinct u_{1}, u_{2}, …, u_{k} ∈ U − {1}, k ⩾ 1, Σ ∩ ⋂ Σ^{u_{k}} is contractible.

The rest of paper is dedicated to the proof of Proposition 3. For simplicity, we deal only with the case k = 1. The same proof works for k > 1.

Lemma 4. (1) For 1 ≠ u ∈ U, T = Σ ∩ Σ^{u} is the subcomplex of Σ consisting of

simplices fixed by u. Moreover, T is a union of chambers.

(2) If Cw ∈ T and v ⩽ w, then Cv ∈ T .

Proof. Let C_{J} = {G_{j}; j ∈ J ⊆ I} ⊆ C and assume C_{J}v_{1}u = C_{J}v_{2} for some v_{1}, v_{2} ∈ W. Then G_{J}v_{1}u = G_{J}v_{2}. As N ∩ G_{J} = N_{J} is the preimage of W_{J} in N, we may choose v_{1} and v_{2} as in Lemma 2. Let l(v_{k}) = r_{k}, k = 1, 2. Without loss assume r_{1} ⩽ r_{2}. Otherwise we consider u^{−1} instead. Set v_{1} = x_{1}x_{2} … x_{r_{1}} with x_{k}’s from S. As sBw ⊆ BwB ∪ BswB for s ∈ S and w ∈ W, we have

v_{1}uv_{2}^{−1} ∈ Bv_{2}^{−1}B ∪ ⋃_{1⩽k_{1}<k_{2}<···<k_{t}⩽r_{1}} Bx_{k_{1}}x_{k_{2}} … x_{k_{t}}v_{2}^{−1}B.

Then as v_{1}uv_{2}^{−1} ∈ G_{J} = BN_{J}B, either v_{1}uv_{2}^{−1} ∈ Bv_{2}^{−1}B ⊆ G_{J} or there are 1 ⩽ k_{1} < k_{2} < ⋯ < k_{t} ⩽ r_{1} such that v_{1}uv_{2}^{−1} ∈ Bx_{k_{1}} … x_{k_{s}}v_{2}^{−1}B ⊆ G_{J}. In the first case, v_{2} ∈ G_{J} and by Lemma 2 v_{2} = 1. Consequently v_{1} ∈ G_{J}v_{2}u^{−1} = G_{J}, and again by Lemma 2 v_{1} = 1. In the second case, set v_{0} = x_{k_{1}} … x_{k_{t}}. Then v_{0} ∈ G_{J}v_{2} with l(v_{0}) ⩽ t ⩽ r_{1} ⩽ r_{2} = l(v_{2}); hence by Lemma 2, all equalities must hold and t = r_{1}, r_{1} = r_{2}, v_{0} = v_{1} = v_{2}. Therefore v_{1}uv_{2}^{−1} ∈ Bv_{0}v_{2}^{−1}B = B, i.e. u ∈ B ∩ B^{v_{1}}. In either case, v_{1} = v_{2}, u ∈ B ∩ B^{v_{1}} and C_{J}v_{1} ⊆ Cv_{1} ⊆ T . Part (1) is proved.

For v ⩽ w ∈ W and Cw ∈ T , assume w = xv with l(w) = l(x) + l(v). Let x = x_{1}x_{2} … x_{r} with r = l(x) and x_{i} ∈ S. For t ⩽ r, set w_{t} = x_{t} … x_{r}v. Now for any s ∈ S and any w ∈ W with sw ⩽ w, we have B ∩ B^{w} ⩽ B^{sw}. Applying the observation to x_{1} and w, we have u ∈ B ∩ B^{w} ∈ B^{w_{2}}, so Cw_{2} ∈ T . Now part (2) follows inductively.

Lemma 5. T is contractible.

Proof. As C ∈ T , T is nonempty. T is connected by Lemma 3 and Lemma 4.2. Recall that w_{0} is the longest element in W. Notice that Cw_{0} ∉ T , as u ∉ B∩B^{w_{0}} = H.

Pick a w̄ ∈ W with Cw̄ ⊆ T such that l(w̄) is maximal. Set

T_{1} = ⋃_{w≠w̄,u∈B∩B^{w}} Cw ⊆ T .

For a simplex σ ∈ T_{1} ∩ Cw̄, choose v ∈ W as in Lemma 3 such that σ ∈ Cv. Then by Lemma 3, v ⩽ w̄. So Cv ∈ T by Lemma 4.2. But σ ∈ T_{1}, and consequently there is a w ∈ W with σ ∈ Cw ∈ T_{1}. This forces v ≠ w̄ and Cv ∈ T_{1}. Therefore by Lemma 3 there is an s ∈ S with v ⩽ sw̄ ⩽ w̄ and σ ∈ Csw̄ ⊆ T_{1}. So σ ⊆ Cw̄∩Csw̄, an (n − 2)-simplex of Cw̄.

On the other hand, for any s ∈ S with sw̄ ⩽ w̄, it is obvious that Csw̄ ∈ T_{1} and Cw̄ ∩ Csw̄ ⊆ T_{1} ∩ Cw̄. We have shown that T_{1} ∩ Cw̄ is the union of (n − 2)- simplices Cw̄ ∩ Csw̄ of Cw̄ with sw̄ ⩽ w̄. As w̄ ≠ w_{0}, there is an s ∈ S with sw̄ > w̄. Therefore there are at most n − 1 (n − 2)-simplices of Cw̄ in T_{1}, and their intersection contains a vertex; hence T_{1} ∩ Cw̄ is contractible. Applying Lemma 1, we conclude that T is homotopy equivalent to T_{1}.

Notice that T_{1} is also a union of chambers and satisfies Lemma 3, part (2). Therefore, repeating the above procedure gives that T is homotopy equivalent to C, the fundamental chamber, hence contractible.

References

[A] M. Aschbacher, Finite Group Theory, Cambridge University Press, Cambridge,1986. MR 95b:20002

[B] Bourbaki, N., Groupes et algebras de Lie, 4, 5, 6, Masson, Paris, 1981. MR 83g:17001

[CL] C.W. Curtis and G. I. Lehrer, A new proof of a theorem of Solomon-Tits, Proc. of AMS v. 82, No. 2 (1982) 154-156. MR 83j:20048

[G] H. Garland, p-adic curvature and the cohomology of discrete subgroups of p-adic groups, Ann. Math.(2) 97 (1973) 375-393. MR 47:8719

[S] L. Solomon, The Steinberg character of a finite group with a BN-pair, Theory of Finite Groups, R. Brauer and C.H. Sah, Eds., Benjamin, New York, 1969, pp. 213-221. MR 40:220

Department of Mathematics, Caltech, Pasadena, California 91125

   E-mail address: chaoku@cco.caltech.edu