L^{2}-cohomology of buildings with fundamental class. ^{1} Jan Dymara ^{2}

Abstract. For an infinite building whose Davis chamber is an orientable, gallery connected pseudomanifold (with boundary), we calculate the top-dimensional L^{2}-Betti number. This gives a complete determination of L^{2}-Betti numbers for 2-dimensional, right-angled hyperbolic buildings.

0. Notation.

Basic facts and definitions concerning buildings and L^{2}-homology can be found in [D], [DO]. We denote the Davis’ realisation of the building by X, the Coxeter system by (W, S) and the Davis complex of W by Σ. The chambers of the Davis complex are in canonical bijection with W; we denote the chamber corresponding to w by D_{w}. We assume throughout that the nerve L (the boundary of the Davis chamber D) is an orientable pseudomanifold of dimension n − 1. We fix an orientation of L and induce an orientation on all n-simplices of D. Then we extend this induced orientation W -equivariantly to all n-simplices of Σ. Also, we orient n-simplices of X pulling back orientations from Σ via a folding map. Notice that if ψ is an n-cycle (or a harmonic n-cochain) on X, and if σ, τ are two n-simplices in X contained in the the same chamber c, then ψ(σ) = ψ(τ). We denote the common value by ψ(c). All the cochains that we will consider will also have this property (we use C^{n}(X) to denote the set of such cochains). For example, if b is a chamber of X we denote by 1_{b} the cochain whose value on simplices contained in b is 1, and whose value on all other simplices is 0. We renormalise the L^{2}-norm, multiplying it by a constant so as to have ∥1_{b}∥ = 1.

We assume that for each s ∈ S and every chamber b of X the number of chambers s-adjacent to b depends only on s and is finite. Then this number q_{[s]} depends in fact only on the conjugacy class [s] of s in W. We denote by q the collection of all the q_{[s]}, and call it the thickness of the building. We also put q^{−1} = (q_{[s]}^{−1})_{s∈S}. For w ∈ W let ℓ_{[s]}(w) be the number of occurrences of generators from a given conjugacy class [s] in a reduced word representing w. Let l(w) = (ℓ_{[s]}(w))_{s∈S}. We define the generating function of

the Coxeter group as follows: W (t) = ∑_{w∈W} t^{l(w)}, where t = (t_{[s]})_{s∈S}, t^{l(w)} = ∏_{[s]} t_{[s]}^{ℓ_{[s]}(w)}. The building X (as well as the Davis complex) is equipped with a W -valued distance function d. Out of this function we produce a family of distances d_{[s]} = ℓ_{[s]} ∘ d. Thus, for two chambers b, c, we get a collection of numbers d(b, c) = (d_{[s]}(b, c))_{s∈S}.

1. The standard n-cycle.

The principal character of this note is the following harmonic n-cochain on X: ϕ_{b}(c) = (−q)^{−d(b,c)}, where b is a chamber in X. Using the b-based folding π_{b} : X → Σ we calculate

(1)  ∥ϕ_{b}∥^{2} = ∑_{c∈X} |ϕ_{b}(c)|^{2} = ∑_{w∈W}(−q)^{−2l(w)}q^{l(w)} = ∑_{w∈W} q^{−l(w)}.

Therefore, ϕ_{b} ∈ L^{2} if and only if q^{−1} is in the convergence region of the power series W (t). Also, if ϕ_{b} ∈ L^{2}, then ∥ϕ_{b}∥^{2} = W (q^{−1}).

Theorem 1

Suppose that (W, S) is an infinite Coxeter system such that the boundary of the Davis chamber is an (n − 1)-dimensional, gallery connected, orientable pseudomanifold. Let X be a building with Weyl group W of thickness q. Then L^{2}H^{n}(X) (the space of square-summable harmonic n-cochains) is non-zero if and only if q^{−1} is in the convergence region of the series W (t).

Proof. (⇒)

Suppose L^{2}H^{n}(X) ≠ 0. Let ψ be a non-zero element of this space. Choose a chamber b such that ψ(b) ≠ 0. Define an n-cochain \overline{ψ} on Σ as follows: \overline{ψ}(c) is the average of the values of ψ on π_{b}^{−1}(c) (recall that π_{b} : X → Σ is the b-based folding). Now put ψ_{b} = \overline{ψ} ∘ π_{b} ∈ C^{n}(X). Notice that harmonicity of ψ implies q^{l(w)}\overline{ψ}(D_{w}) = −q^{l(ws)}\overline{ψ}(D_{ws}) for all s ∈ S, w ∈ W. It follows that \overline{ψ}(D_{w}) = (−q)^{−l(w)}, which gives ψ_{b} = ψ(b)ϕ_{b}. However, by Schwartz’s inequality, ∥ψ_{b}∥ ≤ ∥ψ∥. Therefore ϕ_{b} ∈ L^{2} and, by (1), q^{−1} is in the convergence region of W (t).

1 AMS Classification 58J22 (20F55 20F65).

2 partially supported by KBN grant 5 P03A 035 20 (⇐) For every chamber b, ϕ_{b} is a non-zero element of L^{2}H^{n}(X). ⋄

2. Symmetric buildings. Now suppose that G ⊆ Aut(X) is a closed subgroup with a BN-pair adapted to X. In particular, B

is the stabilizer in G of some chamber in X which we henceforth denote b. One can use G to measure von Neumann dimension dim_{G} of L^{2}H^{i}(X). We denote this dimension by L^{2}b^{i}(X) (it turns out that it does not depend on the choice of G).

Theorem 2 Under all the assumptions of Theorem 1, and the symmetry assumption stated above, we have:

(a) if q^{−1} is in the convergence region of W (t), then L^{2}b^{n}(X) = \frac{1}{W (q^{−1})};

(b) if q^{−1} is not in the convergence region of W (t), then L^{2}b^{n}(X) = 0. Proof. Part (b) follows from Proposition 1. Suppose then that q^{−1} is in the convergence region of W (t).

To find L^{2}b^{n}(X) we need to embed the G-module L^{2}H^{n}(X) into L^{2}(G). Recall that G/B can be identified with the set of all chambers in X. Using this identification and the quotient map p : G → G/B we can pull cochains on X back to G. Equip G with the Haar measure μ, normalised so that μ(B) = 1 (G is unimodular, cf [DJ]). Then the above pull back gives an isometric G-equivariant embedding p^{∗} : L^{2}C^{n}(X) → L^{2}(G).

We need to find the orthogonal projection P : L^{2}(G) → p^{∗}(L^{2}H^{n}(X)). First, the orthogonal projection L^{2}(G) → p^{∗}(L^{2}C^{n}(X)) is given by the right convolution ∗1_{B}. Put Φ = ϕ_{b}∘p ∈ L^{2}(G), and let Φ_{gB} = Φ∘g^{−1}. We have to understand the restricted projection P : p^{∗}(L^{2}C^{n}(X)) → p^{∗}(L^{2}H^{n}(X)).

We will need the following: Lemma In L^{2}C^{n}(X), the orthogonal projection of 1_{b} on L^{2}H^{n}(X) is αϕ_{b}, where α = W (q^{−1})^{−1}. Proof. Since the cochain 1_{b} is B invariant, so is its projection on the B-invariant space L^{2}H^{n}(X). But

all B-invariant harmonic n-cochains are of the form αϕ_{b}.

The coefficient α can be characterised as the unique number for which ⟨ϕ_{b}, 1_{b} − αϕ_{b}⟩ = 0. This gives α = ⟨ϕ_{b}, ϕ_{b}⟩^{−1}, and the Lemma follows due to (1). ⋄(Lemma)

Every function f ∈ p^{∗}(L^{2}C^{n}(X)) can be written as ∑_{g∈G/B} a_{g}1_{gB}. The Lemma gives P (1_{B}) = αΦ; by symmetry, we deduce P (1_{gB}) = αΦ_{gB}. Therefore

(2)  P (f)(u) = ∑_{g∈G/B} a_{g}P (1_{gB})(u) = ∑_{g∈G/B} a_{g}αΦ_{gB}(u) = ∑_{g∈G/B} a_{g}αΦ(g^{−1}u)

On the other hand

(3) \begin{aligned}(f ∗ Φ)(u) & = ∫_{G} f(uh^{−1})Φ(h)dh = ∑_{g∈G/B} a_{g} ∫_{G} 1_{gB}(uh^{−1})Φ(h)dh = ∑_{g∈G/B} a_{g} ∫_{G} 1_{u^{−1}gB}(h^{−1})Φ(h)dh \\ & = ∑_{g∈G/B} a_{g} ∫_{G} 1_{Bg^{−1}u}(h)Φ(h)dh = ∑_{g∈G/B} a_{g}Φ(g^{−1}u) \end{aligned}

Thus α(3) gives the same as (2).

Composing the two projections L^{2}(G) → p^{∗}(L^{2}C^{n}(X)) → p^{∗}(L^{2}H^{n}(X)) we get P = ∗(1_{B} ∗αΦ) = ∗αΦ. The second assertion goes as follows:

L^{2}b^{n}(X) = dim_{G} L^{2}H^{n}(X) = Tr_{G}P = αΦ(e) = α = W (q^{−1})^{−1}. ⋄

3. Hyperbolic right-angled buildings of dimension 2. Let X = X_{p,q} be the right-angled 2-dimensional hyperbolic building, with chamber a p-gon, and thick-

ness q + 1 along every edge (cf [B]). Because of the uniform thickness, we replace each t_{[s]} by t; then W (t) reduces to a one variable power series W (t). Both of these series represent rational functions, which were calculated by Bourdon [B]. In particular,

(4)  \frac{1}{W (t)} = 1 − \frac{p}{1 + t} + \frac{p}{(1 + t)^{2}} ,

so that

(5)  W (t) = \frac{(1 + t)^{2}}{1 + (2 − p)t + t^{2}} .

The smallest (in absolute value) zero of the denominator is \frac{1}{2}(p − 2 − \sqrt{p^{2} − 4p}) = 2(p − 2 + \sqrt{p^{2} − 4p})^{−1}; it lies between (p − 2)^{−1} and (p − 3)^{−1}. This gives the following:

Lemma For q ≥ p − 2 the series (1) converges, while for q < p − 2 it does not. ⋄ We choose G as in section 2 (eg G = Aut(X)), and we want to find the L^{2}-Betti numbers of X. We have L^{2}b^{0}(X) = 0: a 0-cocycle is constant and the set of vertices of X is infinite, so that no 0-cocycle

is in L^{2}. Standard arguments show that L^{2}-Euler characteristic L^{2}χ(X) = L^{2}b^{0}(X) − L^{2}b^{1}(X) + L^{2}b^{2}(X) coincides with the Euler characteristic of the orbifold quotient X/G; the latter is 1 − \frac{p}{1+q} + \frac{p}{(1+q)^{2}} = \frac{1}{W (q)}. Therefore −L^{2}b^{1}(X) + L^{2}b^{2}(X) = \frac{1}{W (q)}, and it suffices to find L^{2}b^{2} to know the other Betti number.

Theorem 3

(a) If q < p − 2, then

L^{2}b^{0}(X) = 0, L^{2}b^{1}(X) = −\frac{1}{W (q)} = −1 + \frac{p}{1 + q} − \frac{p}{(1 + q)^{2}} , L^{2}b^{2}(X) = 0.

(b) If q ≥ p − 2, then

L^{2}b^{0}(X) = 0, L^{2}b^{1}(X) = 0, L^{2}b^{2}(X) = \frac{1}{W (q)} = 1 − \frac{p}{1 + q} + \frac{p}{(1 + q)^{2}} .

Proof. (a) By the Lemma, the series (1) diverges. Therefore, by Proposition 2(b), L^{2}b^{2}(X) = 0. Hence, L^{2}b^{1}(X) = χ(X) = \frac{1}{W (q)}.

(b) By the Lemma and Proposition 2(a), L^{2}b^{2}(X) = \frac{1}{W (q^{−1})}. However, it is easy to check using (4) or (5) that W (q^{−1}) = W (q) (this is no accident, cf [CD]). It follows that L^{2}b^{2}(X) = \frac{1}{W (q)} = χ(X), which implies L^{2}b^{1}(X) = 0. ⋄

Remarks

1. The methods of this note are inspired by Borel’s proof of irreducibility of the Steinberg representation, cf [Bor].

2. We have not touched the question of unreduced cohomology; the L^{2}-cohomology of X is known to be reduced (and the L^{2}-Betti numbers are known) when all q_{[s]} are sufficiently large, cf [DJ].

3. In [BP] Bourdon and Pajot determined the L^{p} cohomology of right-angled hyperbolic buildings using more powerful methods.

References

[Bor] Borel A., Admissible representations of a semi–simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35(1976), 233–259.

[B] Bourdon M., Sur les immeubles fuchsiens et leur type de quasi–isométrie, Ergodic Theory Dynam. Systems 20(2000), no. 2, 343–364.

[BP] Bourdon M., Pajot H., Cohomologie L^{p} et Espaces de Besov, to appear in J. Reine Angew. Math.

[CD] Charney R., Davis M., Reciprocity of growth functions of Coxeter groups, Geom. Dedicata 39 (1991), no. 3, 373–378.

[D] Davis M., Buildings are CAT(0), in Geometry and cohomology in group theory (Durham, 1994), 108–123, London Math. Soc. Lecture Note Ser., 252, Cambridge Univ. Press, Cambridge, 1998.

[DO] Davis M., Okun B., Vanishing theorems and conjectures for the L^{2}-homology of right-angled Coxeter groups, Geom. Topol. 5(2001), 7–74.

[DJ] Dymara J., Januszkiewicz T., Cohomology of buildings and of their automorphism groups, Invent. math. 150(2002) 3, 579–627.

Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.

dymara@math.uni.wroc.pl