Cohomology of buildings and S–arithmetic groups over function fields

Helmut Behr Johann Wolfgang Goethe–Universität Frankfurt am Main

March 2006

Dedicated to Jean-Pierre Serre on the occasion of his eightieth birthday

1 Introduction

This paper is a supplement to the article “Cohomologie d’ immeubles et de groupes S–arithméthiques” by A. Borel and J.P. Serre from 1976 (see [BS]). They deal with S–arithmetic subgroups Γ of reductive algebraic groups G in the case of number fields and discrete cocompact groups, so the general case for function fields F ([F : 𝔽_{q}(t)] < ∞, q = p^{r}) is missing. For a torsion–free group Γ they show that Γ is of type FL, which implies, that Γ is also of type FP_{∞}. For this purpose they compute the cohomology of the spherical Tits–building Y_{v} of the group G(F_{v}), where F_{v} ist the completion of F with respect to v ∈ S, providing Y_{v} with the F_{v}– analytic topology. Using Y_{v} as a boundary at infinity of the Bruhat–Tits–building X_{v} of G(F_{v}), thereby compactifying X_{v}, they can also compute the cohomology with compact supports of X = ∏_{v∈S} X_{v} and prove, that it vanishes in all dimensions, except the top dimension d, which is the sum ∑_{v∈S} dim X_{v} (in the number field case S contains also archimedean places and X_{v} is then the corresponding symmetric space with corners); in dimension d the cohomology with coefficients in ℤ is free.

On the other hand, U. Stuhler considers in his paper “Homological properties of certain arithmetic groups in the function field case” (see [St]) the groups Γ = PGL_{2}(O_{S}) for O_{S} ⊂ F (S finite) and shows in the second part, that the “finiteness length” of Γ is bounded, more precisely, that Γ is not of type FP_{|S|}. Following a remark of J.P. Serre he uses the spectral sequence for the cohomology groups of stabilizers Γ_{σ} (σ a simplex in X/Γ) with coefficients in the finite field 𝔽_{p} and deduces that the vector spaces H^{r}(X/Γ; H^{s}(Γ_{σ}; 𝔽_{p})) are finite–dimensional for 0 ≤ r ≤ |S|, 0 ≤ s < |S|, but definitively not for r = 0, s = |S|. As a consequence H^{|S|}(Γ; 𝔽_{p}) has also infinite dimension, thus Γ cannot be of type FP_{|S|}. For this result Stuhler employs a filtration of X/Γ and computes step by step.

We shall now combine the methods of these two papers: We observe first, that Borel–Serre’s computation also works for certain subcomplexes of Y_{v} and X_{v} which are finite modulo a stabilizer–group Γ_{σ} (σ ∈ X), if we assume that Γ_{σ} has only p–torsion and moreover consists only of elements contained in the unipotent radical U of a minimal F –parabolic group of G. This can be established by changing from Γ to a (congruence) subgroup Γ_{0} of finite index: the existence of Γ_{0} follows from reduction theory in the formulation of Harder. In this way we obtain that the cohomology groups of Γ_{σ} vanish with exception of the top dimension, where we have a description as locally constant functions on Γ_{σ} ⊂ U(F) — a restriction of the results of Borel– Serre. Consequently Stuhler’s spectral sequence degenerates to the isomorphism H^{d}(Γ; 𝔽_{p}) ≃ H^{0}(X/Γ; H^{d}(Γ_{σ}; 𝔽_{p})) and it remains to prove, that this space is infinite–dimensional. This may again be done by a filtration of X/Γ, but it suffices to find an infinite sequence of vertices σ, where Γ_{σ} becomes bigger and defines new elements of H^{0}(X/Γ, H^{d}(Γ_{σ}; 𝔽_{p})). Thus we can at first generalize Borel–Serre’s theorem on the cohomology of S–arithmetic groups Γ with coefficients in ℤ[Γ] to the function field case, when Γ has only p–torsion and secondly prove that all S–arithmetic subgroups Γ of almost simple groups G, defined over F cannot be of type FP_{d}, where d is the sum of the local ranks of G over the fields F_{v} for v ∈ S. (The problem for reductive groups can be reduced to this special case: See [B3], 2.6 c)

Another proof of the last theorem was recently given by K.U. Bux and K. Wortman (see [BW]).

It is conjectured, that these groups are of type FP_{d−1}, which has been shown in special cases or with additional assumptions on the growth of F with respect to the rank: see the precise statements at the end of this paper.

I would like to thank Robert Bieri, Bernd Schulz and Ulrich Stuhler for helpful discussions and critical remarks.

2 Cohomology of spherical buildings

In order to fix the notations and to make clear that Borel–Serre’s computations also work for certain subcomplexes of spherical buildings we have to report [BS], §1 and §2.

Let k be a non–archimedean local field, G a connected semi–simple k-group of k–rank l and Y the Tits–building of G(k). It is well known by the Solomon–Tits–theorem, that Y has the homotopy–type of a bouquet of (l-1)–spheres — with respect to its simplicial topology. The first step in [BS] is to provide Y with the analytic topology, induced by the valuation on k and to prove an analogue, by computing the Alexander–Spanier–cohomology. 2.1 Denote by P a minimal k–parabolic subgroup of G, by T a maximal k–split torus of P, by Δ the basis of simple roots on T, which defines P and by S the associated set of reflections in the Weyl group W of P, W = N(T)/Z(T). We may identify both Δ and S with the index set ℕ_{l} = {1, …, l}, then P_{I} is the parabolic subgroup generated by P and Z(T_{I}), T_{I} the kernel of all a_{i} ∈ Δ_{I} for I ⊆ ℕ_{l}, such that Δ \ Δ_{I} is the set of simple roots of the semi–simple factor of P_{I}; we have P_{∅} = P, P_{N_{l}} = G and J ⊂ I ⇔ P_{J} ⊂ P_{I}. Now let C be the closed chamber of Y, fixed by P (k) and C_{I} its face fixed by P_{I} (for proper subsets I of ℕ_{l}), then Y can be described as G(k)/P (k) × C with identifications. Borel–Serre provide G(k)/P (k) = (G/P)(k) with the k–analytic topology, so it is compact, C with its simplicial topology and Y by the quotient topology, defined by G(k) × C ⟶ G(k)/P (k) × C ⟶^{λ} Y. 2.2 In the next step we use the Bruhat–decomposition G(k) = ∐_{w∈W} P (k)wP (k), which can be made unique by the refinement G(k) = ∐_{w∈W} U_{w}(k)wP (k), where U_{w} is a connected subgroup of the unipotent radical U of P; for the only element w_{0} of maximal length in W one has U_{w_{0}} = U. Thus C(w) := P (k)wP (k)/P (k) is a principal homogenous space over U_{w}(k), therefore isomorphic to k^{d_{w}} with d_{w} = dim U_{w}, especially C(w_{0}) ≃ k^{d_{0}}, d_{0} = dim U. Using this decomposition and enumeration of the Weyl group W = {w_{i}|1 ≤ i ≤ N}, compatible with the length function (i.e. l(w_{i}) ≤ l(w_{j}) for i < j) Borel–Serre construct a filtration of the building as follows: For any w_{m} ∈ W define I_{m} := {i ∈ ℕ_{l}|l(w_{m}s_{i}) > l(w_{m})} for s_{i} ∈ S, thus I_{1} = ℕ_{l} (for w_{1} = id), I_{N} = ∅ and I_{m} ∉ {ℕ_{l}, ∅} for 1 < i < N and setting L_{m} := ⋃_{I⊂I_{m}} C∘_{I} (open faces, in particular C∘_{∅} = C∘), L′_{m} := ⋃_{I⊄I_{m}} C_{I}, we see that w_{m}L′_{m} is the union of all codimension– 1–faces of w_{m}C, which also belong to a chamber which has smaller distance to C than w_{m}C. With E_{m} := ⋃_{1≤i≤m} C(w_{i}), Y_{m} := λ(E_{m} × C) we obtain a filtration of Y = ⋃_{1≤i≤N} Y_{i} with Y_{i} ⊂ Y_{j} for i < j and

Lemma 1: a) Y_{m}\Y_{m−1} = ∐_{I⊂I_{m}} (∏_{I} C(w_{m}) × C∘_{I}), such that Y_{m}\Y_{m−1} is homeomorphic to C(w_{m}) × L_{m}.

b) H̃^{i}(Y; M) = H_{c}^{i}(Y \ Y_{N−1}; M). (M is a ℤ–module, H̃^{∗} the reduced cohomology and H_{c}^{∗} cohomology with compact supports.)

c) H_{c}^{i}(Y \ Y_{N−1}; M) = H_{c}^{l−1}(C∘; M) ⊗ H_{c}^{i−(l−1)}(C(w_{N}); M) which is 0 for all i ≠ l − 1, so H^{i}(Y; M) vanishes for all i ≠ l − 1 and H̃^{l−1}(Y; M) ≃ ℤ ⊗ H_{c}^{0}(C(w_{N}); M).

Remark. It is enough to prove this lemma and the following proposition for M = ℤ: The results for ℤ show that the cohomology modules are 0 or free; therefore the universal coefficient theorem gives the result for a ℤ-module M by tensoring (cf. [Br1], 0.8). Moreover this is also true for the set of locally constant functions with compact support over a totally disconnected space X: C_{c}^{∞}(X; M) = C_{c}^{∞}(X; ℤ) ⊗_{ℤ} M (see [BS], lemma 2.2).

Proof. b) For 1 < m < N we have I_{m} ∉ {∅, ℕ_{l}}, which implies that L′_{m} is contractible, as is C = L_{m} ∪^{.} L′_{m} and the exact cohomology sequence for C mod L′_{m} gives H_{c}^{∗}(L_{m}; ℤ) = 0 and with a) and Künneth’s formula H_{c}^{∗}(Y_{m} \ Y_{m−1}; ℤ) = 0. Now Y_{1} = C is contractible and by induction on m we get b).

c) We already know that Y \ Y_{N−1} = Y_{N} \ Y_{N−1} = C(w_{N}) × C∘, and C∘ is isomorphic to ℝ^{l−1}, oriented by the enumeration of its codimension–1–faces, thus H_{c}^{i}(C∘; ℤ) = 0 for i ≠ l − 1 and H_{c}^{l−1}(C∘; ℤ) = ℤ. The Künneth–formula implies H_{c}^{i}(Y \ Y_{N−1}; ℤ) = H_{c}^{l−1}(C∘; ℤ) ⊗ H_{c}^{i−(l−1)}(C(w_{N}); ℤ), but C(w_{N}) = P (k)w_{N}P (k)/P (k) ≃ k^{d_{0}} is a totally disconnected space, therefore 0–dimensional and has zero cohomology in all dimensions but 0.

2.3 It remains to describe the cohomology for the top dimension. The building Y is the union of all apartments containing C; call this set A and A_{0} the standard apartment, fixed by Z(T) ⊂ P, so A_{0} ∈ A. The correspondence g ↦ gA_{0} defines a bijection between P (k)/Z(T)(k) ≃ U(k) and A, and also C(w_{N}) = U(k)w_{N}P (k)/P (k) ≈ U(k) is in 1–1–correspondence with A: consider C(w_{N}) as the set of chambers in Y, which are opposite to C: A ⟷ U(k) ⟷ C(w_{N}).

Part c) of the lemma also shows that H̃^{l−1}(Y; M) = H_{c}^{0}(A; M) is isomorphic to C_{c}^{∞}(A; M). This isomorphism has an explicit description: The Coxeter–complex Σ_{A} of each A ∈ A is homeomorphic to the sphere 𝕊^{l−1} with orientation, given by the enumeration (Σ_{A} is a labelled complex). The cycle ∑_{w∈W} (−1)^{l(w)}wC defines for l ≥ 2 a class [A] in the homology group H_{l−1}(Y; M) — the “fundamental class of the oriented sphere”. A given element h ∈ H_{c}^{l−1}(Y; M) has by restriction to H_{c}^{l−1}(A; M) a value h([A]) and by definition of cohomology h([A]) is uniquely determined by its value on the opposite chamber uw_{N}C for u ∈ U(k), if A = uA_{0}. So we can summarize as

Proposition 1: H̃^{i}(Y; M) = \{ \begin{matrix}0 & for i ≠ l − 1 \\ H_{c}^{0}(A; M) ≃ C_{c}^{∞}(U(k); M) & for i = l − 1 \end{matrix}

Analyzing the proof above, it is obvious, that it also works for subcomplexes Y′ of Y, which are unions of apartments containing a fixed chamber C; especially if Y′ = ⋃_{u∈U′} uA_{0}, where U′ is a subgroup of U(k).

Corollary: If U′ is a subgroup of U(k), U the unipotent radical of the minimal k–parabolic group P of a semi–simple algebraic k–group G, k a local nonarchimedean field, C the chamber of the Tits–building fixed by P, A_{0} an apartment containing C and Y′ = ⋃_{u∈U′} uA_{0}, then

H̃_{c}^{i}(Y′; M) ≃ \{ \begin{matrix}0 & for i ≠ l − 1 \\ C_{c}^{∞}(U′(k); M) & for i = l − 1 \end{matrix}

3 Cohomology of affine buildings

3.1 Let X be the affine Bruhat–Tits–building, defined by the group G and the valuation on k. We may suppose that G is semi–simple and simply connected (cf. [BS], 4.1), so G is a direct product of almost simple groups G_{j} (1 ≤ j ≤ m) of k–rank > 0 and a simply connected group G_{0} of k–rank 0, such that G_{0}(k) is compact. Then the double Tits–system has also a product structure, the Tits–building Y is the join of the buildings Y_{j} of G_{j} and X is a polysimplicial complex.

The apartments of X are in 1–1–correspondence with those of Y; denote by A_{0} the standard apartment in X, stabilized by N(T) (T ⊂ P as in section 2) and choose an origin O in A_{0} as a special point and a sector (simplicial cone) D with vertex O and direction C, C the standard chamber in Y, fixed by P (for buildings cf. [BT], [Br2] and [R]).

A main part of [BS] contains the construction of a compactification of X by adding Y as a boundary at infinity, such that the induced compactification of any apartment of X is the classical one for an affine space by the sphere of half–lines. Z := X ⨿ Y is shown to have a topology, which makes Z compact and contractible and the action of G(k) on Z continuous. The contraction of X to O is geodesic, which means along half–lines in an apartment. 3.2 In this section we are interested in finite groups Γ_{x}, stabilizing a vertex x ∈ D ⊆ A_{0} and contained in U(k), U = rad P; as we already know (see 2.3), these elements define bijectively apartments A of X, such that \overline{A} contains C in Y and moreover fix pointwise the sector D_{x} with direction C and vertex x (if we assume that x is special — which can be done by changing from x to a neighbour x′ with Γ_{x} ⊂ Γ_{x′} — D_{x} is given by {x′ ∈ A_{0}|α(x′) ≥ α(x)∀α ∈ Δ}).

We consider the subcomplex X′ of X, defined by X′ = ⋃_{γ∈Γ_{x}} γA_{0} and denote by Z′ := X′ ∪ ∂X′, where Y′ := ∂X′ ist the boundary of X′ in Z, so Z′ is compact. We now remove a collar from Z′, in order to get a complex, which is finite modulo Γ_{x}: This can be done by retraction along half–lines with vertex x and — since Γ_{x} is finite — in such a way that different apartments γA_{0} remain different. We obtain a complex X′_{c}, contained in X, homoemorphic to Z′, whose boundary Y′_{c} is homeomorphic to ∂X′ and X′_{c}∩D_{x} is still contained in all apartements γA_{0} for γ ∈ Γ_{x}. Observe that the sectors γD (D with vertex O and ∂D = C) do not coincide, but all have the sector D_{x} in common, which has the same dimension and so D_{x} has the same set of directions as all γD, which implies that ∂γD = ∂D_{x} = C for all γ. For any apartment γA_{0} the intersection γA_{0} ∩ Y′_{c} is homoemorphic to γA_{0} ∩ Y, which is a spherical Coxeter–complex and all these complexes contain the chamber C.

Therefore we can apply the Borel–Serre–method for computing the cohomology of Y′_{c} in the same way as for Y and use the corollary of proposition 1 (notice that Y is the join Y_{1} ∗ ⋯ ∗ Y_{m} and one has the formula dim Y = ∑_{j=1}^{m}(dim Y_{j} + 1) − 1 = ∑_{j=1}^{m} dim X_{j} − 1 = ∑_{j=1}^{m} rk_{k} G_{j} − 1, and l = ∑_{j=1}^{m} rk_{k} G_{j} = rk G), so we get

Lemma 2: \begin{aligned}H̃_{c}^{i}(Y′_{c}; M) & = 0 for i ≠ l − 1, \\ H̃_{c}^{l−1}(Y′_{c}; M) & ≃ C_{c}^{∞}(Γ_{x}; M). \end{aligned}

3.3 The exact sequence of cohomology for X′_{c} modulo Y′_{c} implies (cf. [BS] thm. 5.6)

Proposition 2: \begin{aligned}H_{c}^{i}(X′_{c}; M) & = 0 for i ≠ l and \\ H_{c}^{l}(X′_{c}; M) & ≃ C_{c}^{∞}(Γ_{x}; M) \end{aligned}

for a ℤ-module M; if M is free, then also H_{c}^{l}(X′_{c}; M) is a free ℤ–module.

4 S-arithmetic groups

4.1 Let F be a function field, i.e. [F : 𝔽_{q}(t)] < ∞, q = p^{m}, p = char F, S a finite non–empty set of places of F, F_{v} the completion of F with respect to v ∈ S, G a connected semi–simple algebraic F –group of rank r > 0, r_{v} the F_{v}–rank of G, L = ∏_{v∈S} G(F_{v}), X_{v} the Bruhat–Tits– building of G(F_{v}) with dimension d_{v} = r_{v}, X = ∏_{v∈S} X_{v} with dim X = ∑_{v∈S} d_{v} =: d.

We consider S–arithmetic subgroups Γ, which are discrete in L. It is well known, that Γ contains a congruence subgroup Γ_{0} of finite index, which has only p–torsion, but we need a little bit more and have to use for this purpose 4.2 (Reduction theory) We use the same notations as in section 2, but with respect to F instead of k: P is a minimal F –parabolic subgroup of G, P = Z(T) ▷◁ U, Δ = {α_{1}, …, α_{r}} the set of simple roots of T, that defines P. We give a list of properties in the version of Harder (see [H1], cf. also [H2], [B2] and for the 1–dimensional case [S2], II.2).

(i) There is a constant c_{1}, such that for any x ∈ X, there exists a minimal P with numerical invariants ν_{i}(P, x) ≥ c_{1} for 1 ≤ i ≤ r ([H1], Satz 2.3.2): “x is reduced with respect to P ”.

(ii) There is a constant c_{2} ≥ c_{1}, such that for x reduced with respect to P and P′ and ν_{i}(P, x) ≥ c_{2} ∀i ∈ I′ ⊆ Δ, P′ is also contained in P_{I} for I = Δ \ I′ and P_{I} is uniquely determined ([H1], Satz 2.3.3): “x is close to P_{I}”.

(iii) For each constant c′ ≥ c_{1} the set

X_{0} := {x ∈ X | c_{1} ≤ ν_{i}(P, x) ≤ c′ for all i ∈ Δ and all P s.t. x is P -reduced}

is compact modulo Γ (cf. [H1], Satz 2.2.2): “compactness criterion”.

(iv) The number of Γ–conjugation classes of parabolic groups is finite ([B2], Satz 8): “finiteness

of class number”.

(v) There exists a constant c_{3} ≥ c_{2} (depending on Γ), such that for a P –reduced x ∈ X one has U_{I}(F_{v}) = (U_{I}(F_{v}) ∩ stab_{G} x) · (U_{I}(F_{v}) ∩ Γ) for all v ∈ S (cf. [H2], 1.4.5): “x is very close to P_{I}, U_{I} = rad P_{I}”.

4.3 ¿From these properties above we deduce the following proposition; for subgroups H ⊆ G we shorten the notation: F_{S} := ∏_{v∈S} F_{v} and H(F_{S}) := ∏_{v∈S} H(F_{v}).

Proposition 3: For each S–arithmetic group Γ ⊂ G(F_{S}), there exists a congruence subgroup Γ_{0}, finitely many F –parabolic subgroups P_{1}, …, P_{h_{0}} of G and a set V of representatives for vertices of X/Γ_{0}, such that for all x ∈ V the stabilizers (Γ_{0})_{x} are contained in one of the groups [R_{u}(P_{j})](F_{S}), R_{u}(P_{j}) the unipotent radical of P_{j}. Moreover, those x ∈ V which are close to P_{j} are contained in an apartment, having P_{j} and a fixed opposite P_{j}^{op} in its boundary at infinity.

Proof. a) X_{0} := {x ∈ X | c_{1} ≤ ν_{i}(P, x) ≤ c_{2} for all i ∈ Δ and all P s.t. x is P -reduced} is compact modulo Γ by (iii), thus there exist finitely many vertices in X, which represent the vertices of X_{0} and have finite stabilizers in Γ. Therefore we can find a congruence subgroup Γ_{1} of Γ, which has trivial stabilizers for all x ∈ X_{0}.

b) Now we consider those x ∈ X, which are close to at least one parabolic group Q (but not to Q′ ⊂ Q), which means that some ν_{i}(P, x) ≥ c_{2} for P ⊆ Q. Up to conjugation with Γ_{1} we may assume that P belongs to a finite set {P_{1}, …, P_{h}} by (iv) and Q = (P_{j})_{I} for some j and I ⊆ Δ_{j}, so there are only finitely many such groups Q. We write Q = H⋉U_{Q} with H = Z(T′), T′ ⊆ T_{j}, the maximal split torus of P_{j} and U_{Q} = R_{u}(Q) ⊆ R_{u}(P_{j}). We want to “split up” Γ_{1}: There is a subgroup Γ_{2} of finite index in Γ_{1}, such that Γ_{2} = (Γ_{1} ∩ H(F_{S}))⋉(Γ_{1} ∩ U_{Q}(F_{S})) has finite index in Γ (cf. [Bo], 1.7 and 8.12, which is also valid for function fields, see [B1]). Now H is either of F –rank 0, if I = ∅ or the numerical invariants ν_{i}(H ∩ P, x) are bounded by c_{2} for all i ∈ Δ \ I = I′ (the set of simple roots of H!), so either H(F_{S})/Γ_{1} ∩ H(F_{S}) is compact or by the same idea as in a) we find a congruence subgroup Γ_{3} of Γ_{2}, whose stabilizer of x ∈ X has a trivial semi–simple component in [Γ_{3} ∩ H(F_{S})], so (Γ_{3})_{x} ⊆ U_{Q}(F_{S}) ⊆ [R_{u}(P_{j})](F_{S}). We can repeat this process for the finite set of groups Q and finally obtain a congruence subgroup Γ_{0} of Γ, whose stabilizers have only unipotent elements; a set of representatives can be found in the unipotent radicals [R_{u}(P_{j})](F_{S}), but now for a bigger set j = 1, …, h_{0} for the Γ_{0}–conjugates of minimal parabolic groups.

At last poperty (v) of 4.2 shows, that we can choose modulo Γ_{0} a fixed opposite group P_{j}^{op} of P_{j} for all j and this implies the second assertion of proposition 3 – by making Γ_{0} smaller and h_{0} bigger if necessary.

5 Cohomology of S-arithmetic groups

5.1 Stuhler uses in [St] the following spectral sequence

(∗)  H^{r}(X/Γ, H^{s}(Γ_{σ}; M)) ⟹ H^{r+s}(Γ; M)

which was introduced by Serre (for a foundation cf. [Br1], VII.5 and VII.7). Stuhler dealt with Γ = SL_{2}(O_{S}), X a product of |S| trees, the Bruhat–Tits–buildings of SL_{2}(F_{v}), v ∈ S and M = 𝔽_{p}. In our context Γ will be a S–arithmetic subgroup of G(F) and X the polysimplicial product of the Bruhat–Tits–buildings X_{v} for G(F_{v}), v ∈ S. 5.2 For the application of the results in section 2 and 3 we have to observe that a minimal parabolic F –subgroup P is in general not minimal over F_{v}: Denote by Q a minimal F_{v}–parabolic group, contained in P for some v ∈ S. We have P = HT ⋉R_{u}(P) with a maximal split F –torus T and a semi–simple group H of F –rank 0; therefore H(F_{S})/H(F_{S}) ∩ Γ is compact. There may be infinitely many Q ⊂ P, but since we are interested in vertices x of a set V of representatives for X/Γ with Γ_{x} ⊂ [R_{u}(P)](F), where P is one of the minimal parabolic F –subgroups of proposition 3, the compactness above tells us, that we have to consider only finitely many Q’s. Moreover R_{u}(P) ⊆ R_{u}(Q) (as groups over F_{v}), but for R_{u}(Q) = R_{u}(P) · U′, the subgroup U′ of H cannot contain unipotent elements of Γ, so [R_{u}(P)](F_{v}) ∩ Γ = [R_{u}(Q)](F_{v}) ∩ Γ.

5.3 The building Y_{v} at infinity of X_{v} has a chamber C_{Q}, fixed by Q(F_{v}) and C_{Q} has a side– simplex C_{P}, fixed by P (F_{v}) (C_{P} = (C_{Q})_{I} for P = Q_{I} in section 2). Consider now a finite stabilizer group Γ_{x} ⊂ [R_{u}(P)](F) for x = (x_{v}) ∈ ∏_{v∈S} X_{v} = X. In any apartment A of X_{v}, containing x_{v}, the group Γ_{x} fixes a cone in X_{v}, which is the union of sectors, which all have the “side–direction” Q in common. The construction of the subcomplexes X′ and X′_{c} in section 3 can now be realized simultaneously in all buildings X_{v} with the same group Γ_{x}, the stabilizer of a vertex x = (x_{v}) ∈ X = ∏_{v∈S} X_{v} in Γ, which is diagonally imbedded in ∏_{v∈S} G(F_{v}). We start with a fixed apartment A_{0} in X_{v} and define X′_{v} = ⋃_{γ∈Γ_{x}} γA_{0}; \overline{A}_{0} ∩Y_{v} may contain several chambers C_{Q} for Q ⊂ P, but the equation [R_{u}(Q)](F_{v}) ∩ Γ = [R_{u}(P)](F_{v}) ∩ Γ shows that X′_{v} depends only on P. On the other hand we can use any such chamber C_{Q} for the computation of the cohomology as in section 2. 5.4 We fix now a vertex x = (x_{v}) in the polysimplicial complex X and denote by Γ_{x} its stabilizer in Γ, assuming that Γ is the congruence subgroup of an S–arithmetic group, which has the properties of proposition 3. With this group Γ_{x} construct the subcomplexes (X′_{v})_{c} as in section 3 for all v ∈ S and define X′_{c} = ∏_{v∈S}(X′_{v})_{c}.

Proposition 2 gives us the cohomology of (X′_{v})_{c} and the Künneth–formula implies

Proposition 4: \begin{aligned}H_{c}^{i}(X′_{c}; M) & = 0 for i ≠ d, \\ H_{c}^{d}(X′_{c}; M) & ≃ ⨂_{v∈S} H_{c}^{d_{v}}((X′_{v})_{c}; M) \end{aligned}

for d = ∑_{v∈S} d_{v}, d_{v} = dim X_{v} = rk_{F_{v}} G and a ℤ-module M.

This proposition allows to deduce the cohomology of the stabilizer groups.

Lemma 3:  H^{i}(Γ_{x}; ℤ[Γ_{x}]) = 0 for i ≠ d, H^{d}(Γ_{x}; ℤ[Γ_{x}]) ≃ C_{c}^{∞}(Γ_{x}; ℤ).

Proof. We use the isomorphism H^{∗}(Γ_{x}; ℤ[Γ_{x}]) ≃ H^{∗}(X′_{c}, ℤ) (see [Br1], VIII.7, ex. 4), which is valid, since X′_{c} is contractible and has only finitely many cells mod Γ_{x} (and of course finite isotropy groups). For the precise description of H^{d}(Γ_{x}; ℤ[Γ_{x}]), we have to remember Lemma 1, which says together with the corollary to proposition 1, that H_{c}^{d_{v}}((X′_{v})_{c}; M) ≃ H̃_{c}^{d_{v}−1}((Y′_{v})_{c}; M) = H_{c}^{d_{v}−1}(C∘_{v}; M) ⊗ H^{0}(Γ_{x}; M), where C∘_{v} denotes the interior of a chamber C_{v} in Y_{v}. For X′_{c} we must consider the chamber C = ∗_{v∈S}C_{v} (join) in Y = ∗_{v∈S}Y_{v} and so we obtain H^{d}(X′_{c}; M) ≃ H_{c}^{d−1}(C∘; M) ⊗ H^{0}(Γ_{x}; M), where the first factor is isomorphic to ℤ (since C∘ ≃ ℝ^{d−1}) and the second one to C_{c}^{∞}(Γ_{x}; M): Set M = ℤ.

Corollary:  a) H^{i}(Γ_{x}; ℤ[Γ]) = 0 for i ≠ d, H^{d}(Γ_{x}; ℤ) ≃ C_{c}^{∞}(Γ_{x}; ℤ) ⊗_{ℤ[Γ_{x}]} ℤ[Γ] ≃ C_{c}^{∞}(Γ_{x}; ℤ[Γ]). b) H^{i}(Γ_{x}; 𝔽_{p}) = 0 for i ≠ d, H^{d}(Γ_{x}; 𝔽_{p}) ≃ C_{c}^{∞}(Γ_{x}; ℤ) ⊗_{ℤ[Γ_{x}]} 𝔽_{p} ≃ C_{c}^{∞}(Γ_{x}; 𝔽_{p}).

Proof. For the second parts see [Br1], VIII.6.8, since Γ_{x} is of type FP and cd Γ_{x} = d.

5.5 For the application of the spectral sequence we have to consider the stabilizers of polysimplices σ, but Γ_{σ} is always the intersection of some Γ_{x} and we notice that the results above remain valid for Γ_{σ}. On the other side we have to replace the trivial Γ_{σ}–module ℤ by the “oriented module” ℤ_{σ}, on which Γ_{σ} acts with a sign. Then it is true, that the spectral sequence H^{r}(X/Γ; H^{s}(Γ_{σ}; ℤ_{σ} ⊗ M)) converges to H^{r+s}(Γ; M) (see [S1] 1.6, especially remark 1). The corollary implies, that the spectral sequences degenerate to isomorphisms, so we obtain Theorem 1: For a S–arithmetic subgroup Γ of G(F), G a semi–simple algebraic F –group with rk_{F} G > 0, there exists a subgroup Γ_{0} of finite index, which has only p–torsion, such that for the Γ_{0}–modules M = ℤ[Γ_{0}] or M = 𝔽_{p} (with trivial Γ_{0}-action) and the Bruhat–Tits–building X = ∏_{v∈S} X_{v} we have

(i) H^{i}(Γ_{0}; M) = 0 for i ≠ d = ∑_{v∈S} rk_{F_{v}} G,

(ii) H^{d}(Γ_{0}; M) ≃ H^{0}(X/Γ_{0}; H^{d}(Γ_{0})_{σ}; M),

in particular H^{d}(Γ_{0}; ℤ[Γ_{0}]) is free. Remark. For rk_{F} G = 0, the “cocompact case”, see [BS], thm. 6.2.

6 Finiteness properties of S-arithmetic groups

6.1 In this section we are interested in the finiteness properties FP_{n} or F_{n} (see [Br1], VIII.5 and [Br2], VII.2). In contrast to the number field case, where all S–arithmetic subgroups of reductive groups are of type F_{∞} (in the general case it is only true for S = S_{∞}), there exist many counter–examples over function fields and a conjecture says, that for almost simple groups G with rk_{F} G > 0 Γ is of type F_{d−1}, but not F_{d}, where again d = ∑_{v∈S} rk_{F_{v}} G. This was proved for Γ = SL_{2}(O_{S}) by Stuhler (see [St]), for the classical cases d = 1 (finite generation) and d = 2 (finite presentation)) (see [B3]), for Γ = SL_{n}(𝔽_{q}[t]) by Abels and Abramenko under the additional assumption, that q is big enough with respect to n and by the second author also for classical almost simple groups over the polynomial ring and with analogous growth conditions (see [A] and [Ab]) and finally the positive result for Chevalley–groups and arithmetic rings with |S| = 1 without such a condition in [B4]. 6.2 There is a necessary cohomological condition for a group Γ to be of type FP_{n}: For a ring R, which is a ℤ–module of finite type, the homology and cohomology groups H_{k}(Γ; R) and H^{k}(Γ; R) have to be finitely generated for 0 ≤ k ≤ n (see [Bi], prop. 2.15). Stuhler used this criterion for R = 𝔽_{p} for the group SL_{2}(O_{S}), to show that it is not of type FP_{|S|} (see [St], section 4). We shall generalize his idea for the congruence subgroup Γ_{0} of Γ (from proposition 3) in almost simple groups G, using

H^{d}(Γ_{0}; 𝔽_{p}) ≃ H^{0}(X/Γ_{0}; H^{d}((Γ_{0})_{σ}; 𝔽_{p})) ≃ H^{0}(X/Γ_{0}; C_{c}^{∞}((Γ_{0})_{σ}; 𝔽_{p}))

(see theorem 1 and the corollary to lemma 3). We prove that the 𝔽_{p}–dimension of this vector– space goes to infinity for an infinite sequence of vertices σ = x in a set V_{P} of representatives for X/Γ_{0} being very close to a fixed parabolic F –subgroup P = P_{j} for some j ∈ {1, …, h_{0}} (see 4.2 (v) and proposition 3). We may also assume that these vertices are contained in poly– apartements on which a fixed maximal F –split torus T_{0} of P acts. Denote by a_{0} the highest F –root on T_{0}, that defines a subgroup U_{0} of the unipotent radical U of P. Consider a_{0} as a linear form with integral values on all these apartements.

By Riemann–Roch U_{0}(x) := U_{0}(F_{S}) ∩ (Γ_{0})_{x} is for x = (x_{v}) ∈ X a 𝔽_{q}–vector–space. If a_{0}(x_{v}) increases by 1 for some v, all others remaining constant, dim_{𝔽_{q}} U_{0,x} increases by d_{v} = dim_{𝔽_{q}} \overline{F}_{v} (\overline{F}_{v} the residue field of F_{v}), if U_{0} is a 1–dimensional root group and otherwise has moreover to be multiplicated by dim_{F} U_{0}(F). Thus it seems reasonable to define the filtration of V_{P} by A_{0}(x) = ∑_{v∈S} d_{v} · a_{0}(x_{v}), A_{0} is Γ–invariant by the product–formula for valuations. The whole set of representatives for X/Γ_{0} consists of a finite complex and finitely many sets, containing the V_{P_{j}}.

6.3 For the computation of H^{0}(X/Γ_{0}; C_{c}^{∞}((Γ_{0})_{x}; 𝔽_{p})) we must satisfy the condition for 0– cocycles, which means concretely that the values of a function on the vertices x and x′ of an edge have to coincide on (Γ_{0})_{x} ∩ (Γ_{0})_{x′}. We have to consider several cases:

a) If rk_{F} G = rk_{F_{v}} G = 1 for all v ∈ S we have a cubic complex X (the prototype is Stuhler’s SL_{2}–example) and U_{0} = U_{a} or U_{0} = U_{2a}. We consider a cube C, whose vertices x and y have maximal resp. minimal A_{0}–value; the neighbours of x′ being x_{i} with a_{0}[(x_{i})_{v_{i}}] = a_{0}[(x)_{v_{i}}] − 1 and a_{0}[(x_{i})_{v_{k}}] = a_{0}[(x)_{v_{k}}] for all k ≠ i, setting S = {v_{1}, …, v_{s}}, i, k ∈ {1, …, s}. Unfortunately U_{0}(x) = ⋃_{i=1}^{s} U_{0}(x_{i}), thus all functions on U_{0}(x) are uniquely determined by their values on the subgroups U_{0}(x_{i}). But functions on the vertices x_{ij}, whose a_{0}- values differ from those on x for two indices by 1 have different extensions to functions on x_{i} and x_{j}. This can be done for all pairs (i, j), which implies that the 𝔽_{q}–dimension of U_{0}(y) is smaller than that of U_{0}(x) and so we obtain more functions on (Γ_{0})_{x} than on (Γ_{0})_{y} and H^{0} becomes larger (cf. [St], prop. 2 and lemma 2). One should observe, that many vertices close to P are Γ–equivalent, since the torus T_{0}(F_{s}) ∩ Γ_{0} acts on these points as a free abelian group of rank |S| − 1. But the function A_{0} is Γ–invariant and obviously A_{0}(y) < A_{0}(x), so x and y are different vertices of V_{P}.

b) If G is a Chevalley–group of rk_{F} G = rk_{F_{v}} G ≥ 2 for all v ∈ S than in a fixed building X_{v} the group U_{0}(x) = U_{0}(F_{v}) ∩ (Γ_{0})_{x} is larger than U_{0}(x′) for all neighbours x′ of x, so also (Γ_{0})_{x} is larger than the union ⋃_{x′}(Γ_{0})_{x′} and we obtain more cochains. For |S| > 1 we use the same technique as in a).

c) If rk_{F} G < rk_{F_{v}} G both situations may occur in X_{v}: U_{0}(x) ⊂ ⋃_{x′} U_{0}(x′) and U_{0}(x) = ⋃_{x′} U_{0}(x′) (where x′ runs over all neighbours of x). Simple examples are groups G of type B_{2,1} (G = SO_{5} with Witt–index 1) or type A_{3,1} (G = SL_{2}(D), D central division algebra of degree 2 over F). For the second case the same trick as in a) works: We have to go one A_{0}–level deeper to a common neighbour y of x′_{1} and x′_{2}. For |S| > 1 use again the procedure of case a). If the values of A_{0} go to infinity we find in any case a sequence of vertices x with increasing 𝔽_{p}–dimension of C_{c}^{∞}((Γ_{0})_{x}; 𝔽_{p}), for which also the 𝔽_{p}–dimension of H^{0} goes to infinity.

6.4 With the criterion of 6.2 we proved, that Γ cannot be of type F_{d} — for a convenient subgroup of finite index, but this does not change the F_{d}–property (see [Br1], VIII.5.1) – thus we have Theorem 2: A S–arithmetic subgroup of an almost simple algebraic group G with rk_{F} G > 0 over a function field F with d = ∑_{v∈S} rk_{F_{v}} G cannot be of type FP_{d} or F_{d}. Remarks. 1. The direct application of the finiteness criterion to semi–simple groups could

provide too large upper bounds, because cohomology groups may vanish if other tensor– factors in Künneth’s formula are zero. Thus for semi–simple groups the bound in theorem 2 is given by the minimal d of its simple factors (assuming G simply connected we have a direct product: see [B3] 2.6 c).

2. There exists another proof of theorem 2 by K.U. Bux and K. Wortman (see [BW]), which does not compute cohomology but constructs a sequence of cycles in homology.

References

[A] H. Abels : Finiteness properties of certain arithmetic groups in the function field case; Israel J. Math 76 (1991), 113–128

[Ab] P. Abramenko : Twin Buildings and Applications to S–Arithmetic Groups; Springer Lecture Notes in Math. vol 1641 (1996)

[B1] H. Behr : Zur starken Approximation in algebraischen Gruppen über globalen Körpern; J. Reine und Angew. Math. 229 (1968), 107–116

[B2] H. Behr : Endliche Erzeugbarkeit arithmetischer Gruppen über Funktionenkörpern; Invent. Math. 7 (1969), 1–32

[B3] H. Behr : Arithmetic groups over function fields I; J. Reine und Angew. Math. 495 (1998), 79–118

[B4] H. Behr : Higher finiteness properties of S–arithmetic groups in the function field case I; in Groups: Topological, Combinatorical, Arithmetic Aspects; London Math. Soc. Lecture Notes 311 (2004), 27–42

[Bi] R. Bieri : Homological dimension of discrete groups; Queen Mary College Math. Notes (1976)

[Bo] A. Borel : Some finiteness properties of adele groups over number fields; Publ. IHES 16 (1963), 5–30

[Br1] K. Brown : Cohomology of groups; Springer GTM 87 (1982)

[Br2] K. Brown : Buildings; Springer (1989)

[BS] A. Borel and J. P. Serre : Cohomologie d’ immeubles et de groupes S–arithmétiques; Topology 15 (1976) 211–232

[BT] F. Bruhat and J. Tits : Groupes réductifs sur un corps local I, II; Publ. Math. IHES 41 (1972), 5–251 and 60 (1984), 5–184

[BW] K.U. Bux and K. Wortman : Finiteness properties of arithmetic groups over function fields; to appear in Invent. Math.

[H1] G. Harder : Minkowskische Reduktionstheorie über Funktionenkörpern; Invent. Math. 7 (1969), 33–54

[H2] G. Harder : Die Kohomologie S–arithmetischer Gruppen über Funktionenkörpern; Invent. Math. 42 (1977), 135–175

[R] M. Ronan : Buildings: Main ideas and applications I, II; Bull. London Math. Soc. 24 (1992), 1–51 and 97–126

[S1] J. P. Serre : Cohomologie des groupes discrets; Annals of Math. Studies 70 (1971), 77–169 [S2] J. P. Serre : Trees; Springer (1980)

[St] U. Stuhler : Homological properties of certain arithmetic groups in the function field case; Invent. Math. 57 (1980), 263–281