Cohomology of buildings and finiteness properties of S-arithmetic groups over function fields.

Helmut Behr, Frankfurt am Main

July 2011

S-arithmetic subgroups Γ of reductive algebraic groups G over number fields are finitely presented and contain a torsion-free subgroup of finite index, which is of type F L (Ragunathan 1968, Borel-Serre 1976), therefore they are of type F P_{∞}, i.e. there exists a projective resolution

P_{m} → P_{m−1} → ⋯ → P_{0} → ℤ → 0

of finitely generated ℤΓ-modules P_{i} for all m, and also of type F_{∞}, i.e. there exists an Eilenberg-MacLane complex K(Γ, 1) with finite m-skeleton for all m (cf. [Br2], VIII).

For function fields F ([F : 𝔽_{q}(t)] < ∞, q = p^{k}, p = charF) however, many counter-examples are known: SL_{2}(𝔽_{q}[t]) is not even finitely generated, i.e. not of type F_{1} (Nagao 1959, Serre 1968), SL_{2}(𝔽_{q}[t, t^{−1}]) and SL_{3}(𝔽_{q}[t]) are finitely generated, but not finitely presented, i.e. of type F_{1}, not F_{2} (Stuhler 1976, Behr 1977); for the S-arithmetic ring O_{S} (S a finite, non-empty set of primes of F), SL_{2}(O_{S}) is of type F_{|S|−1}, but not F_{|S|} (Stuhler 1980), SL_{n}(𝔽_{q}[t]) is of type F_{n−2} but not F_{n−1} as long as q ≥ 2^{n−2} (Abels 1989) or q ≥ \binom{n−2}{[\frac{n−2}{2}]} (Abramenko 1987) and similar results hold for absolutely almost simple 𝔽_{q}-groups G and Γ = G(𝔽_{q}[t]) (Abramenko 1994), the positive part was proved without restriction on q for Chevalley groups G, Γ = G(O_{S}), |S| = 1 (Behr 2004).

All these results provided evidence for the following conjecture: If G is an absolutely almost simple algebraic group, defined over F with F -rank r > 0 (“isotropic group”), r_{v} = rank_{F_{v}} G for the completion F_{v} of F (v ∈ S, S finite, S ≠ ∅), Γ S-arithmetic subgroup (discrete in G_{S} = Π_{v∈S}G(F_{v})), then Γ is of type F_{d−1}, but not of type F_{d} for d = ∑_{v∈S} r_{v}.

This conjecture was proved for the classical properties finite generation (iff d ≥ 2, Behr 1969, Keller 1980) and finite presentation (iff d ≥ 3: Behr 1998) and the negative part for arbitrary d (Bux-Wortman 2007). Moreover for anisotropic G (i.e. rk_{F} G = 0) it was known (Serre 1971), that Γ is of type F_{∞}.

The positive part is now proved in a preprint by Bux – Gramlich – Witzel ([BGW], 2011). This paper gives a completely different and relatively short proof, using two old results on cohomology, which has the advantage not to need such precise local informations, which are necessary in (almost) all proofs of the results mentioned above, and, also in [BGW], which use filtrations of buildings, defined in very clever ways.

In 1976 Borel-Serre also computed the cohomology of spherical and affine buildings over non-archimedean local fields. If X_{v} is the Bruhat-Tits-building of G(F_{v}) (G a semi-simple F -group, v ∈ S), then the product X = Π_{v∈S}X_{v} is a contractible polysimplicial complex. X gives rise to a chain-complex C = (C_{n})_{n∈ℕ} with ℤΓmodules C_{n}, generated by polysimplices, but these are (in general) not projective nor finitely generated. Borel-Serre proved that the reduced cohomology with compact support H̃_{c}^{i}(X; M) for a ℤΓ− module M vanishes in all dimensions, except for the top-dimension d = ∑_{v∈S} d_{v}, d_{v} = dimX_{v} = r_{v} = rank_{F_{v}} G. For H_{c}^{d}(X; M) they gave an explicit description by locally-constant functions on unipotent groups. In section 1 we present a short version of their results.

On the other hand, K. Brown found in 1975 a very interesting cohomological criterion for finiteness properties. In his proof he constructed for a Γ-complex C_{n} of projective modules another complex C′_{n} with the same homology, but finitely generated Γ-modules. His assumptions are not all valid in our case and so we cannot use his general arguments. In section 2 we use his construction, but we must be more explicit, on the other side our situation is more special: The crucial point is Borel-Serre’s vanishing result for cohomology with compact supports; in some sense, this substitutes the notion of being essentially trivial for filtrations (see [Br3],2). We obtain a partial resolution of ℤ by finitely generated free Γ-modules up to dimension d − 1, so Γ is of type F P_{d−1}. By this method we cannot prove type F_{d−1}, but this is implied together with finite presentation for d ≥ 3 ([B3]).

In section 3 we give a proof for the negative part, which is based on the same ideas as the proof in [BW], but the computations of Borel-Serre provide a more natural construction of cycles, that also illustrates the geometry of Bruhat-Tits-buildings: we find arbitrary big spherical holes in a mod Γ compact subcomplex X_{0} of X, which only become boundaries of cones, arbitrary far from X_{0}.

I am very grateful for interesting comments by Herbert Abels, Robert Bieri, Kenneth Brown, Kai-Uwe Bux, Ulrich Stuhler and especially to Wolf-Hanno Rehn for pointing out several mistakes.

1 Borel-Serre: Cohomology of buildings and S-arithmetic groups (see [BS])

1.1 Spherical or Tits buildings

Let k be a non-archimedean local field, G a connected semi-simple k-group of rank l ≥ 1 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. Borel-Serre provide Y with the analytic topology, induced by the valuation on k and prove an analogue for the Alexander-Spanier-cohomology (cf. [Sp1] or [Sp2]).

Denote by P a minimal k-parabolic subgroup of G and by C the closed chamber of Y fixed by P (k). Then Y can be described as G(k)/P (k)×C with identifications. The homogeneous space G(k)/P (k) = (G/P)(k) is compact for the k-analytic topology, C carries the simplicial topology and Y gets the quotient-topology: Y_{t}.

We have P (k) = Z_{G}(T)(k) · U(k), where T is a maximal k-split torus of G, Z_{G} its centralizer and U the unipotent radical of P, W the corresponding Weyl group. The Bruhat-decomposition G(k) = P (k)W P (k) = U(k)W P (k) gives a more concrete description of Y. Especially for the longest element w_{0} of W this product-decomposition is unique; therefore we have a 1-1-correspondence between U(k) and the set C_{P} of all chambers opposite to C, defined by u ↦ u · w_{0}C. Since two opposite chambers determine an apartment (cf. [BT], 4), the set A_{P} of all apartments containing C is also in 1-1-correspondence with U(k) and thus inherits a k-analytic structure from that of U(k). A_{P} = {uA_{o}|u ∈ U(k)}, where A_{o} is the apartment, defined by the opposite pair (C, w_{0}C) and fixed by Z_{G}(T)(k) = P (k)∩ w_{0}P (k)w_{0}^{−1}. In this setting Borel-Serre can compute the (reduced) Alexander-Spanier cohomology H^{∗}(Y_{t}; M) for a ℤ-module M, using the group C_{c}^{∞}(A_{P}; M) of locally constant functions with compact support on A_{P} or U(k):

Proposition 1. (=[BS], thm. 2.6)

(i) H^{i}(Y_{t}; M) = 0 for i ≠ l − 1

(ii) H̃^{l−1}(Y_{t}; M) ≃ C_{c}^{∞}(U(k); M)

1.2 Affine or Bruhat-Tits-buildings

The affine (or euclidean) building X of G(k) (for a non-archimedean local field k) is more conveniently defined for the simply-connected covering G̃ of G in order to obtain X as a product of the buildings X_{j} for the almost simple factors G_{j} of G̃ – so it is a polysimplicial complex (see [BS], 4). An important part of this paper ([BS], 5) consists of the construction of a compactification of X by adding Y as a boundary at infinity. Thereby the direct sum Z = X ⨿ Y becomes a contractible compact space Z_{t}, inducing the natural topology on X and Y_{t} on Y.

Using the long exact cohomology sequence for Z_{t} mod Y_{t}

⋯ → H̃^{i}(Z_{t}; M) → H̃^{i}(Y_{t}; M) → H_{c}^{i+1}(X; M) → H^{i+1}(Z_{t}; M) → …

where M is a module over a ring R and H_{c}^{∗} denotes the cohomology with compact supports and moreover by the vanishing of H̃^{∗}(Z_{t}; M) we can transfer proposition 1 to

Proposition 2. (=[BS], thm. 5.6)

(i) H_{c}^{i}(X; M) = 0 for i ≠ l

(ii) H_{c}^{l}(X; M) ≃ \{ \begin{matrix}H^{l−1}(Y_{t}; M) & if & l ≥ 1 \\ M & if & l = 0 \end{matrix}

in particular for R = ℤ and l ≥ 1:

H_{c}^{l}(X; M) ≃ C_{c}^{∞}(U(k); M)

Remark: A function f ∈ C_{c}^{∞}(U(k); M) has compact support and is locally constant, so there exists a finite union of open subsets of U(k), such that f is constant on each of them. This union corresponds to a neighbourhood of Y_{t} in Z_{t} and f is determined on its compact complement on X.

1.3 S-arithmetic groups over function fields

Let F be a function field (i.e. [F : 𝔽_{q}(t)] < ∞, q = p^{m}, p = char F) with a finite non-empty set S of places of F and F_{v} the completion of F with respect to v ∈ S.

G denotes a connected semi-simple algebraic F -group of rank r, r_{v} := rank_{F_{v}} G (v ∈ S), G_{S} := Π_{v∈S}G(F_{v}); X = Π_{v∈S}X_{v} with Bruhat-Tits-buildings X_{v} of G(F_{v}), dim X_{v} = d_{v} = r_{v} and d = dim X = ∑_{v∈S} d_{v}. Finally Γ is a S-arithmetic subgroup, discrete in G_{S}.

G is called isotropic if r > 0 and anisotropic if r = 0. It is well known from reduction theory (“Godement’s compactness criterion”; cf. [H] or [B1]) that X/Γ is compact iff G is anisotropic.

For this cocompact case Borel-Serre prove (thm. 6.2 in [BS])

Proposition 3. A S-arithmetic subgroup Γ of an anisotropic connected semisimple group G over a function field F is finitely presented and of type F P_{∞} and also of type F_{∞}.

Remarks:

a) More precisely they show that Γ has a torsion free subgroup Γ_{0} of finite index, which is of type F L and so of type F (P)_{∞}, thus Γ inherits this properties.

b) Moreover H^{i}(Γ_{0}; ℤΓ_{0}) ≃ \{ \begin{matrix}0 & for i ≠ d \\ H_{c}^{d}(X; ℤ), & thus free \end{matrix}

c) In the “number-field-case” (over a number field K instead of F) all these results are valid for arbitrary S-arithmetic groups, i.e. also for isotropic G.

In the function-field-case Bux and Wortman proved in [BW], that Γ can be of type F_{∞} only for anisotropic G: They give a bound for the “finiteness length” (max{n : Γ has type F_{n}}) for isotropic groups. To obtain a sharp bound, one should restrict to absolutely almost simple groups: A simply connected semi-simple group G is the direct product of its almost simple factors G_{i} and so is Γ = ∏ Γ_{i}, Γ_{i} in G_{i}. For instance Γ can only be finitely generated if all Γ_{i} are so. In this situation they show, that Γ cannot be of type F P_{d} (so not F_{d}).

1.4 Isotropic groups

Now assume that G is an isotropic, absolutely almost simple F -group. So there exists a minimal F -parabolic subgroup P with unipotent radical U. For each v ∈ S choose a minimal F_{v}-parabolic group Q_{v}, contained in P with unipotent radical U_{v}, such that U_{v}(F_{v}) ⊇ U(F_{v}) ⊃ U(F).

Set \overline{U}_{S} := ∏_{v∈S} U_{v}(F_{v}) ⊇ U_{S} := ∏_{v∈S} U(F_{v}) ⊃ U(F) (the last inclusion by diagonal embedding) and Γ ⊂ G(F) is S-arithmetic. By propositions 1 and 2 we have the isomorphisms for an arbitrary ℤ−module M

H_{c}^{d_{v}}(X_{v}; M) ≃ H_{c}^{d_{v}−1}(Y_{v}; M) ≃ C_{c}^{∞}(U_{v}(F_{v}); M)

and H_{c}^{i}(X_{v}; M) = 0 for i < d_{v}, so for X = ∏_{v∈S} X_{v} we obtain by Künneth’s formula

Theorem 1. (cf. [BS] , 6.6)

a) H_{c}^{i}(X; M) = 0 for i < d = ∑_{v∈S} dim X_{v};

b) H_{c}^{d}(X; M) = ⨂_{v∈S} H_{c}^{d_{v}}(X_{v}; M) ≃ C_{c}^{∞}(\overline{U}_{S}; M)

2 Construction of a F P_{d−1}-resolution for Γ

The Bruhat-Tits-building X provides an augmented chain-complex C = (C_{n})_{−1≤n≤d} with ℤΓ− modules C_{n}, generated by the n−dimensional polysimplices of X for n ≥ 0 and C_{−1} = ℤ, C_{n} →^{∂n} C_{n−1}. Since X is a contractible space, C has trivial reduced homology.

Following K. Brown (see [Br1]) we shall construct inductively a projective — or even free — resolution of ℤ by defining chain-complexes C′(k) = (C′(k)_{n})_{−1≤n≤d−1} with finitely generated ℤΓ−modules C′(k)_{n}, where C′(k)_{n} = C′(k − 1)_{n} for n ≤ k − 1 and C′(k)_{n} = 0 for n > k, beginning with C′_{−1} = ℤ (derivation ∂′_{n} : C′_{n} → C′_{n−1}).

Moreover we define chain-maps f_{k} : C′(k) → C, starting with f_{−1} = id_{ℤ} and f_{k} an extension of f_{k−1}. Thereby we obtain finally a subcomplex C′ of C, whose support in X is compact modulo Γ.

We consider the mapping-cones C′′(k) for f_{k−1}, given by C′′(k)_{n} := C_{n}⊕C′(k−1)_{n−1}, C′′(k)_{−1} = ℤ and ∂′′_{n}(c, c′) = (∂_{n}c−f_{n−1}(c′), −∂′_{n−1}(c′)), ∂′_{−1} = 0.

2.1 Homology and the beginning of induction

There is a short exact sequence

0 → C → C′′ → ΣC′ → 0, (ΣC′)_{n} := C′_{n−1},

giving rise to a long exact sequence for homology

(1) … → H_{n}(C) → H_{n}(C′′(k)) → H_{n−1}(C′(k − 1)) → H_{n−1}(C) → …

Denote by X_{o} the set of vertices of X, then we have

C′′(0)_{0} = C_{0}(X) ⊕ ℤ = {(Σz_{i}x_{i}, z′) | x_{i} ∈ X_{0}; z_{i}, z′ ∈ ℤ}

and with ∂_{0}(Σz_{i}x_{i}) = Σz_{i} (augmentation-map) we obtain ∂′′_{0}((Σz_{i}x_{i}, z′)) = 0 ⇔ Σz_{i} = z′. Furthermore C′′(0)_{1} = C_{1}(X) ⊕ {0}; H_{0}(C) = 0 implies

Z_{0}(C) = B_{0}(C) ≃ B_{0}(C′′(0)) = {Σz_{i}x_{i}, 0) | Σz_{i} = 0}

Choose now a base point x_{0} ∈ X_{0}, then (Σz_{i}(x_{0} − x_{i}), 0) ∈ B_{0}(C′′(0)) and we see that

H_{0}(C′′(0)) ≃ {((Σz_{i}) · x_{0}, (Σz_{i}))} = {(zx_{0}, z) | z ∈ ℤ} ≃ ℤ

We can give evidence of its ℤΓ−module-structure:

H_{0}(C′′(0)) ≃ {(Σz_{γ}(γx_{0}), Σz_{γ}) | z_{γ} ∈ ℤ, γ ∈ Γ}_{/{(Σz_{γ}(γx_{0}),0)|Σz_{γ}=0}}

Using this description, we can lift the augmentation-map ϵ : ℤΓ → ℤ to a ℤΓ−homomorphism φ_{0} of ℤΓ into Z_{0}(C′′(0)), defined by φ_{0}(Σz_{γ}γ) := (Σz_{γ}(γx_{0}), Σz_{γ}); so φ_{0} surjects onto H_{0}(C′′(0)). Z_{0}(C′′(0)) can be viewed as a fiber-product C_{0} ×_{ℤ} C′_{−1}, given by the maps ∂_{0} and f_{−1}:

\xymatrix{ℤΓ \ar@{-->}[drr]^{∂′_{0}} \ar[dr]^{φ_{0}} \ar@{-->}[ddr]_{f_{0}} \\  & Z_{0}(C′′(0)) \ar[d] \ar[r] & C′_{−1} = ℤ \ar[d]^{f_{−1}} \\  & C_{0} \ar[r]_{∂_{0}} & ℤ}

According to this diagram we define

C′_{0} := ℤΓ, f_{0} : C′_{0} → C_{0} by f_{0}(Σz_{γ}γ) : = Σz_{γ}(γx_{0})
∂′_{0} : C′_{0} → C′_{−1} by ∂′_{0}(Σz_{γ}γ) : = Σz_{γ}

As a consequence we have to set C′′(1)_{1} = C_{1} ⊕C′_{0} and C′′(1)_{0} = C_{0} ⊕ℤ, C′′(1)_{n} = C_{n} for n > 1. We confirm that

∂′′_{0} ∘ ∂′′_{1}(c_{1}, c′_{0}) = ∂′′_{0}(∂_{1}(c_{1}) − f_{0}(c′_{0}), −∂′_{0}(c′_{0}))
= (∂_{0} ∘ ∂_{1}(c_{1}) − ∂_{0} ∘ f_{0}(c′_{0}) − f_{−1} ∘ (−∂′_{0}(c′_{0}), −∂′_{1} ∘ (−∂′_{0})(c′_{0}))
= 0, (since ∂_{0} ∘ f_{0} = f_{−1} ∘ ∂′_{0})

Moreover ∂′′_{1}(0, −c′_{0}) = ( − f_{0}(−c′_{0})), −∂′_{0}(−c′_{0})) = (Σz_{γ}(γx_{0}), Σz_{γ}) for c′_{0} = Σz_{γ}γ, which means that ∂′′_{1} : C′′(1)_{1} → C′′(1)_{0} is surjective on H_{0}(C′′(0)), thus H_{0}(C′′(1)) = 0.

Observe that f_{0}(C_{0}) = f_{0}(ℤΓ) = ℤ · (Γx_{0}), whose support is the subcomplex Γ · x_{0} =: X′_{0} of X.

For the next step consider H_{0}(C′(0)) = Z_{0}(C′(0)) = {Σz_{γ} ·γ | Σz_{γ} = 0} =: IΓ, the augmentation ideal of ℤΓ. It is well known that IΓ is a finitely generated ℤΓ−module iff Γ is a finitely generated group. Now Γ is a S−arithmetic subgroup of G(F), G an absolutely almost simple algebraic F −group — what we assume from now on. Therefore Γ is finitely generated iff the sum of local ranks d ≥ 2 (see [B1] or [B3]); so we find a free module (ℤΓ)^{r_{1}} =: C′(1)_{1}, which surjects on IΓ = H_{0}(C′(0)). By sequence (1) H_{0}(C′(0)) is isomorphic to H_{1}(C′′(1)), because H_{1}(C) = H_{0}(C) = 0. We can lift the surjection of C′(1)_{1} onto H_{1}(C′′(1)) to Z_{1}(C′′(1)), which is by definition a fiberproduct C_{1} ×_{C_{0}} Z_{0}(C′_{0})with respect to the maps ∂_{1} and f_{0}. As above in the diagram we get the maps f_{1} : C′(1)_{1} → C_{1} and ∂′_{1} : C′(1)_{1} → C′(1)_{0} = C′(0)_{0}, more concretely:

\xymatrix{c′_{1} \ar@{-->}[drr]^{∂′_{1}} \ar[dr] \ar@{-->}[ddr]_{f_{1}} \\  & (c_{1}, c′_{0}) \ar[d] \ar[r] & c′_{0} \ar[d]^{f_{0}} & with ∂′_{0}(c′_{0}) = 0 \\  & c_{1} \ar[r]_{∂_{1}} & ∂_{1}(c_{1}) = f_{0}(c′_{0})}

Let us point out, that c_{1} is a 1−chain in C_{1}(X), whose boundary is contained in f_{0}(C′_{0}). Since C′(1)_{1} is finitely generated, we get finitely many elements c_{1}^{1}, …, c_{1}^{r_{1}}, whose supports are paths p_{1}, …, p_{r_{1}} in X, which generate a Γ−subcomplex X′_{1} of X with X′_{1}/Γ compact.

For each c′_{0} ∈ Z_{0}(C′(0)) there exists c_{1} ∈ C_{1} with ∂_{1}c_{1} = f_{0}(c′_{0}), since H_{0}(C) = 0 and also c′_{1} ∈ C′(1)_{1} with ∂′_{1}(c′_{1}) = c′_{0} : H_{0}(C′(1)) = 0. Alternatively we could use C′′(2)_{2} := C_{2} ⊕ C′(1)_{1} and compute ∂′′_{2}, proving that H_{1}(C′′(2)) = 0, which implies by (1) H_{0}(C′(1)) = 0.

2.2 Some examples

1. Γ = SL_{2}(𝔽_{q}[t]) is not finitely generated, due to Nagao-Serre (see [S], II. 1.6); its Bruhat-Tits-building is a tree X, which has a half-line H as a fundamental domain mod Γ. H has vertices x_{i}(i ∈ ℕ_{0}) with stabilizers Γ_{i} = stab_{Γ}x_{i}, where

Γ_{0} = SL_{2}(𝔽_{q}), Γ_{i} = ( \begin{matrix}𝔽_{q}^{∗} & 𝔽_{q}[t]_{i} \\ 0 & 𝔽_{q}^{∗} \end{matrix} ) with 𝔽_{q}[t]_{i} = \{p ∈ 𝔽_{q}[t] | deg p ≤ i\} for

i > 0.

We have X′_{0} = Γ · x_{0}, which contains γ_{i}x_{0} for γ_{i} ∈ Γ_{i}⧹Γ_{i−1} : the shortest path p_{i} in X, connecting x_{0} with γ_{i}x_{0} must contain the vertex x_{i}. Thus the complex X′_{1}, generated by the p_{i} is not compact mod Γ. If c′_{i} = γ_{i}x_{0} − x_{0} ∈ C(0)′_{0} and c_{i} is the chain corresponding to p_{i} in C′_{1} we have (c_{i}, c′_{i}) ∈ Z_{1}(C′′(1)), which shows, that H_{1}(C′′(1)) cannot be a finitely generated ℤΓ− module — just as H_{0}(C′(0)).

2. Γ = SL_{3}(𝔽_{q}[t]) is finitely generated: it is easy to see, that E = SL_{3}(𝔽_{q}) ∪ {e_{12}(p), e_{23}(q) | p, q ∈ 𝔽_{q}[t]_{1}} is a set of generators. A standard apartment A of its Bruhat-Tits-building X is a triangulated plane and a fundamental domain for X mod Γ is given by a cone C in A with vertex x_{0} and angle \frac{π}{3}. Let △_{0} be the triangle with vertex x_{0} and contained in C, s.th. for each γ ∈ E we have γ△_{0} ∩ △_{0} ≠ ϕ (γ fixes at least one vertex of △_{0}). This implies, that every vertex γx_{0} is connected with x_{0} by a path p, that projects into △_{0}, so p is contained in Γ△_{0} =: X′_{1}, which means X′_{1}/Γ is compact. In the language of chains: For each c′ ∈ Z_{0}(C′(0)) with f_{0}(c′) ∈ C_{0}(X′_{0}) ⊂ C_{0}(X), X′_{0} = Γ · x_{0} we find c ∈ C_{1}(X′_{1}) ⊂ C_{1}(X) s. th. (c, c′) ∈ Z_{1}(C′′(1)). But these pairs generate H_{1}(C′′(1)), since (c, 0) ∈ Z_{1}(C′′(1)) is a boundary by H_{1}(X) = 0 and for (c_{1}, c′), (c_{2}, c′) ∈ Z_{1}(C′′(1)) we have (c_{1}, c′) − (c_{2}, c′) = (c_{1} − c_{2}, 0) ∈ B_{1}(C′′(1)). Conclusion: H_{1}(C′′(1)) is a finitely generated ℤΓ− module, because the elements c′ = γx_{0} − x_{0} with γ ∈ E generate Z_{0}(C′(0)).

On the other side SL_{3}(𝔽_{q}[t]) is not finitely presented: this is shown in [B2] by constructing an infinite series of closed paths in X′_{1} (or 1−cycles c′_{n} ∈ C′_{1}(1)_{1}), which cannot be contracted in ΓC_{n} where C_{n}(n ∈ ℕ) are compact subsets of C with ⋃_{n} C_{n} = C. In the same way as in example 1 we obtain

elements (c_{n}, c′_{n}) in Z_{2}(C′′(2)), which cannot be contained in a finitely generated ℤΓ−module and of course are inequivalent mod B_{2}(C′′(2)) — s.th. H_{2}(C′′(2)) is not finitely generated as a ℤΓ−module and the supports of all 2−chains c_{n} cannot be contained in a complex X′_{2} with X′_{2}/Γ compact.

3. Γ = SL_{2}(𝔽_{q}[t, t^{−1}]) is finitely generated by the set E = SL_{2}(𝔽_{q}) ∪ {e_{12}(p) |

p ∈ 𝔽_{q}[t]_{1}\}∪\{( \begin{matrix}t & 0 \\ 0 & t^{−1} \end{matrix} )\} . An apartment A of X = X_{1}×X_{2} ( the buildings

X_{1} of SL_{2}(𝔽_{q}[t]) and X_{2} of SL_{2}(𝔽_{q}[t^{−1}])) is a plane, divided into squares. Choose x_{0} ∈ A with stab_{Γ}x_{0} = SL_{2}(𝔽_{q}), then there exists a finite union Q of squares, containing x_{0}, s.th. γQ ∩ Q ≠ ϕ for all γ ∈ E. Then each γx_{0} ∈ Γx_{0} =: X′_{0} is connected with x_{0} by a path in ΓQ =: X′_{1}, so X′_{1}/Γ is compact and we conclude as in example 2, that H_{1}(C′′(1)) is generated by elements (c, c′) with c ∈ C_{1}(X′_{1}), so it is a finitely generated ℤΓ−module.

On the other hand, Γ is not finitely presented, which was shown in [St1] in a similar way as for example 2, s. th. H_{2}(C′′(2)) cannot be finitely generated.

4. G = SL_{2} × SL_{2} is semi-simple, but not almost simple and Γ = Γ_{1} × Γ_{2} = SL_{2}(𝔽_{q}[t]) × SL_{2}(𝔽_{q}[t]) is not finitely generated, since the Γ_{i} are not and H_{0}(C′(0)) = IΓ_{1} × IΓ_{2}, isomorphic to H_{1}(C′′(1)), is not a finitely generated ℤΓ−module — although the sum d of local ranks is 2!

2.3 Cohomology

The examples show that in order to construct a resolution with free ℤΓ− modules and a mod Γ finite subcomplex of X, one should prove that H_{k}(C′′(k)) is a finitely generated ℤΓ− module for k ≤ d − 1. Brown’s construction in [Br1] uses cohomology for this purpose, but he assumes, that the given modules of chains are projective, which is not true in our case - so we need more special arguments. We shall apply Borel-Serre’s theorem (thm. 1) on cohomology with compact supports. They refer to Alexander-Spanier-cohomology, but for the polysimplicial complex X it can also be described by homomorphisms on the ℤΓ−modules C_{n}(X) of chains (cf [Sp1], section 19 or [Sp2], 6.9). A compact support then consists of finitely many chains in C_{n}(X), just as for the more abstract complexes C′_{n} and C′′_{n}.

Recall that an element of H^{n}(X; M) with an arbitrary coefficient-module M is given by an abelian group homomorphism φ on C_{n}(X), which must be zero on the module B_{n}(X) of boundaries (since δ_{n}φ := φ ∘ ∂_{n+1}) and can be modified by δ_{n−1}ψ = ψ ∘ ∂_{n} with ψ : C_{n−1}(X) → M. The action of Γ on φ is given by (γφ)(c) = γ[φ(γ^{−1}c)]. We are interested in finitely generated ℤΓ−modules of φ′s; sometimes it is more convenient, assuming that the supports of generators contain only one element in each Γ-orbit, to consider the Γ-orbit {γφ | γ ∈ Γ} as a Γ−homomorphism φ̃ with φ̃(γc) = γφ̃(c). The finite generation of the module of φ′s is equivalent to say, that the module of Γ−homomorphisms φ̃ has compact (or finite) support mod Γ.

Now we construct the complexes C′ and C′′ by induction and also the subcomplex X′ of X, where X′/Γ is finite and f(C′) = C(X′). We start induction with

X′_{0} := Γ · x_{0}(x_{0} ∈ X),  C′(0)_{0} := ℤΓ,  f_{0}(∑ z_{γ}γ) = ∑ z_{γ}(γx_{0}).

Suppose now k ⩾ 1 (I should point out, that we don’t need the first step, described in 2.1, which used homology and the theorem on finite generation of Γ). Define the Γ−homomorphism φ

φ: C′′(k)_{k} = C_{k} × C′(k − 1)_{k−1} ⟶ C_{k−1}, given by φ(c, c′) = ∂_{k}(c),

of course the support of φ is not compact mod Γ. Since C′(k − 1)_{k} = 0, we have B_{k}(C′′(k)) = B_{k}(C) = Z_{k}(C), so φ vanishes on B_{k}(C′′(k)) and defines an element of H^{k}(C′′(k); C_{k−1}). We can modify φ in its cohomology class by (−π_{1} ∘ ∂′′_{k}), π_{1} the projection of C′′(k − 1) to its first component C_{k−1}, thus we have

π_{1} ∘ ∂′′_{k}(c, c′) = π_{1}(∂_{k}c − f_{k−1}c′, ∂′_{k−1}c′) = ∂_{k}c − f_{k−1}c′.

The restriction of the class of φ to homology is a well-defined element \overline{φ} ∈ Hom_{Γ}(H_{k}(C′′(k)); C_{k−1}).

(Remark: In the more comfortable situation of [Br 1] one even has an isomorphism between H^{k}(C′′(k); M) and Hom_{Γ}(H_{k}(C′′(k)); M) for arbitrary M.) We have

\overline{φ}(c, c′) = ∂_{k}(c) = f_{k−1}(c′).

The image of \overline{φ} is H_{k−1}(f_{k−1}(C′(k − 1))) since ∂_{k−1}(f_{k−1}c′) = ∂_{k−1} ∘ ∂_{k}(c) = 0 and because H_{k−1}(C) = 0 there exists for each cycle f_{k−1}(c′) an element c ∈ C_{k} with ∂_{k}(c) = f_{k−1}(c′).

For isotropic groups we know, that X/Γ is not compact, so we have to consider a filtration of X, i.e. an ascending sequence of subcomplexes X(m) with X(m)/Γ compact and U_{m}X(m) = X. Denote by φ_{m} the restriction of φ to C(X(m))_{k} × C′(k − 1)_{k−1}, then φ_{m} has compact support mod Γ (the second component C′(k − 1)_{k−1} is a finitely generated ℤΓ− module by induction) and defines an element of H_{c}^{k}(C′′(k); C_{k−1}); its restriction \overline{φ}_{m} may be non-trivial only for a finitely generated submodule of H_{k}(C′′(k)).

For cohomology there exists also a long exact sequence

(2) … → H_{c}^{n−1}(C; M) → H_{c}^{n−1}(C′(k−1); M) → H_{c}^{n}(C′′(k); M) → H_{c}^{n}(C; M) → …

for arbitrary coefficient-modules M. For n ≤ d − 1 Borel-Serre’s result says, that H_{c}^{n}(C; M) = 0 which implies H_{c}^{k}(C′′(k); M) ≃ H_{c}^{k−1}(C′(k − 1); M) for k ≤ d − 1. By induction C′(k − 1) is a complex of finitely generated free ℤΓ− modules and therefore its cohomology is also finitely generated, which means that the module of the corresponding Γ−homomorphisms has compact support modulo Γ− and by the isomorphism this ist also true for H_{c}^{k}(C′′(k); M).

We apply this conclusion to the Γ−homomorphisms φ_{m} with M = C_{k−1} and its cohomology classes: They vanish outside some C(X(m_{0}))_{k}. In particular we have \overline{φ}_{m} = \overline{φ} for all m ⩾ m_{0} for the restrictions to the homology H_{k}(C′′(k)). Observe that the support of ∂_{k}: C_{k} → C_{k−1} is of course not compact mod Γ, but there may be different elements c_{1} and c_{2} ∈ C(k), s.th. (c_{1}, c′) and (c_{2}, c′) define the same homology class, which means (c_{1}, c′)−(c_{2}, c′) ∈ B_{k}(C′′(k)) ⇒ ∂_{k}c_{1} = f_{k−1}c′ = ∂_{k}c_{2}.

We summarize: For each c′ ∈ Z_{k−1}(C(k − 1)) = H_{k−1}(C′(k − 1)), c′ ≠ 0 with ∂′_{k−1}c′ = 0 and ∂_{k−1}(f_{k−1}c′) = f_{k−2}(∂′_{k−1}c′) = 0 we find c ∈ C_{k} with ∂_{k}c = f_{k−1}c′, because H_{k}(C) = 0. Since \overline{φ}(c, c′) = ∂_{k}c = 0 for c ∉ C(X(m_{0})) we must have c ∈ C(X(m_{0})) for all c′ ∈ H_{k−1}(C′(k − 1)). X(m_{0})/Γ is compact, so there exists a set X′_{k} of k−dimensional polysimplices, X′_{k} ⊂ X(m_{0}) and C(X′_{k})/Γ finite, which supports the first component of H_{k}(C′′(k)).

We choose a (minimal) finite set {c_{1}, …, c_{r}} of generators for C(X′_{k}). There may exist several elements c′ with f_{k−1}c′ = ∂_{k}c_{i}, but we may assume by induction that they are congruent mod Γ (even mod the finite stabilizer Γ_{0} of ∂_{k}c_{i}). Thus we can also choose c′_{i} ∈ C′(k − 1)_{k−1}, s.th. \{\overline{(c_{i}, c′_{i})} | i = 1, …, r\} is a finite set of generators for the Γ−module H_{k}(C′′(k)), where \overline{(c, c′)} is the class of (c, c′) ∈ Z_{k}(C′′(k)).

Now we can define C′(k)_{k} := (ZΓ)^{r} and get the following diagram for

φ_{k}(∑_{i=1}^{r} z_{γ_{i}} · γ_{i}) := ∑_{1}^{r} z_{γ_{i}} · γ_{i}(c_{i}, c′_{i}) ∈ Z_{k}(C′′(k)).

\xymatrix{(ℤΓ)^{r} \ar@{-->}[drr]^{∂′_{k}} \ar[dr]^{φ_{k}} \ar@{-->}[ddr]_{f_{k}} \\  & Z_{k}(C′′(k)) \ar[d]^{pr_{1}} \ar[r]^{pr_{2}} & Z_{k−1}(C′(k − 1)) ⊂ C′(k − 1)_{k−1} \ar[d]^{f_{k−1}} \\  & C(X(m_{0}))_{k} \ar[r]_{∂_{k}} & C(X(m_{0}))_{k−1}}

∩ ∩
C_{k} C_{k−1}

thus defining ∂′_{k} : C′(k)_{k} → C′(k)_{k−1} and

f_{k} : C′(k)_{k} → C_{k}.

Moreover we have C′′(k + 1)_{k+1} := C_{k+1} ⊕ C′(k)_{k} and the k−dimensional

subcomplex X′(k) := ⋃_{j=1}^{k} X′_{j}, adding X′_{k}.

We also saw, that ∂′_{k} surjects on Z_{k−1}(C′(k − 1)), which means H_{k−1}(C′(k)) = 0, so we obtained a resolution of ℤ by finitely generated free ℤΓ−modules up to dimension k.

Remarks

1. Under the assumptions of [Br1], one can even show, that H^{k}(C; M) ≃ Hom_{ℤΓ}(H_{k}(C); M) and if the cohomology of C preserves direct limits for the coefficient modules, Brown constructs the chain-complex C′ with finitely generated ℤΓ−modules.

2. For the semi-simple group G of example 4 the construction above should not work, although for d = 2 the sequence (2) implies that H_{c}^{1}(C′′(1), C_{0}) is finitely generated.

Here we have X′_{0} = ℤ(Γ_{1}×Γ_{2})·(x_{0}^{1}, x_{0}^{2}) and the shortest path between (x_{0}^{1}, x_{0}^{2}) and (γ_{i}^{1}x_{0}^{1}, x_{0}^{2}) must contain the vertex (x_{i}^{1}, x_{0}^{2}) (notations as in example 1 with upper index 1 or 2) s.th. the distance between x_{0}^{1} and γ_{i}^{1}x_{0}^{1} goes to infinity with i−analogous in the second component. There exists a closed path in X = X_{1} × X_{2} with the following vertices:

(x_{0}^{1}, x_{0}^{2}) → (x_{i}^{1}, x_{0}^{2}) → (γ_{i}^{1}x_{0}^{1}, x_{0}^{2}) → (γ_{i}^{1}x_{0}^{1}, x_{j}^{2}) → (x_{i}^{1}, x_{j}^{2}) → (x_{0}^{1}, x_{j}^{2}) → (x_{0}^{1}, x_{0}^{2}). This path p (or chains, whose support is p) is a boundary in X(d ≥ 2), therefore a homomorphism from cohomology has to vanish on p, e.g. ∂_{1}, which should define the isomorphism on homology. Bur for fixed x_{i}^{1} we get x_{2}^{i} arbitrary far from x_{2}^{0}, which means, that ∂_{1} cannot be restricted to compact supports.

We summarize our results in the following

Proposition 4. a) There exists a partial resolution of Z with free ℤΓ−modules of finite rank: C′_{d−1} → C′_{d−2} → … → C′_{0} → Z

b) There exists a (d − 1)−dimensional subcomplex X′ = X′(d − 1) of X with H_{k}(X′) = 0 for k < d − 1.

Both properties imply the following finiteness theorem: Cf. for the first [Br2], VIII. 4.3 and for the second [Br3], 1.1 (observing that stabilizers in Γ of cells in X are finite).

Theorem 2. A S−arithmetic subgroup Γ of an absolutely almost simple algebraic group G, defined over a function field F with rank_{F} G > 0 and d = ∑_{r∈S} rank_{F_{v}}G is

of type F P_{d−1}.

Remarks:

1. For semi-simple groups we obtain type F P_{d′−1}, where d′ is the minimal d for the simple factors.

2. Our (co)homological method cannot prove that Γ is also of type F_{d−1}; but this is true, since finite presentability was shown for d = 3 (see [B3]: unfortunately this proof is case-by-case and lengthy and part II of it was not published, but exists!): cf [Br2], VIII. 7.

3 Spherical holes in X imply, that Γ is not of type F P_{d}.

This theorem cannot be deduced immediately from Borel-Serre’s result on the top-cohomology of X. But their computations provide a natural construction of cycles. We prove that the codim 1−homology is not essentially trivial and use then Brown’s criterion. An important tool is the existence of a section of apartments, which is compact modulo Γ — already used in [BW].

3.1 Homology and cohomology in the top-dimension

For v ∈ S let Y_{v} be the Tits-building and A an apartment of Y_{v} in A_{P} (cf. 1.1). A is an oriented sphere and its homology H_{l−1}(A; ℤ) is generated by the cycle ∑_{w∈W}(−1)^{lw}wC in the fundamental class [A]. An element h of the cohomology

H̃^{l−1}(A; ℤ) can be restricted as a function to homology: The values h([A]) establish the isomorphism H̃^{l−1}(Y_{v,t}; ℤ) ≃ C_{c}^{∞}(A_{p}; ℤ), the set of locally constant functions with compact support on A_{p} (see [BS], 2.5-2.6). Roughly speaking, the cohomology may be understood as the characteristic function of supports for homology.

Recall that the standard apartment A_{0} is determined by a pair (C, C^{op}) of opposite chambers and each A ∈ A_{p} by a pair (C, uC^{op}) with u ∈ U(F_{v}), giving the 1−1−correspondence between A_{p} and U(F_{v}), s.th. H̃^{l−1}(Y_{v,t}; ℤ) ≃ C_{c}^{∞}(U(F_{v}); ℤ). For the spherical join Y of the Y_{v}(v ∈ S) we have to consider the product U_{S} =

∏_{v∈S} U(F_{v}) and obtain H̃^{d−1}(Y_{t}; ℤ) ≃ C_{c}^{∞}(U_{S}; ℤ).

The transition to affine buildings with boundary is based on the sequence (cf. 1.2)

0 → Y_{t} → Z_{t} → X → 0

and its long exact sequence for cohomology shows that H^{l−1}(Y_{t}; ℤ) ≃ H_{c}^{l}(X; ℤ), since the cohomology of Z_{t} vanishes.

This isomorphism can be explained as follows: The (l − 1)−cycles in Y_{t} are given by fundamental classes [\overline{A}] of apartments \overline{A} ∈ A_{P} in Y_{t} and \overline{A} bounds an apartment A of X. By retraction of A ∪ \overline{A} towards a center z_{A} ∈ A we can remove a collar and can consider [\overline{A}] as a (l − 1)-cycle in X; there it is the boundary of a l−chain c_{A}. In the top-domension c_{A} is unique, its support is a cone with vertex z_{A} and edges, whose directions represent vertices in Y_{t}, defining chambers in \overline{A}. Observe that the support of c_{A} in X is compact. In the other direction: A given support of a l−chain c_{A} may be extended in different ways towards Y_{t}. But since the cohomology is locally constant, their extension is constant for a neighbourhood in Y_{t}.

3.2 Example: Γ = SL_{2}(0_{S})

It should be useful to see the proof at first for a simple example without too many technicalities. We choose the first example, for which the theorem was proved in 1980 by U. Stuhler, computing the 𝔽_{p}−cohomology of Γ.

For SL_{2}(0_{S}) the Bruhat-Tits-building X is the product of s =| S | trees X_{v}, its apartments are ℝ^{s} with a right-angled complex. The apartments of the boundary Y are spherical joins, its chambers (s − 1)-simplices. A pair of opposite minimal parabolic subgroups is (P^{+}, P^{−}), the groups of upper and lower triangular matrices with Levi-decompositions P^{+} = T ⋉ U^{+}, P^{−} = T ⋉ U^{−}. The standard apartments A_{0} and \overline{A}_{0} are determined by the opposite chambers C and C^{−}, fixed by P_{S}^{+} = ∏_{v∈S} P^{+}(F_{v}) and P_{S}^{−}. The torus T_{S} acts on A_{0}; if we choose an origin o = (o_{v})

in A_{0}, we get A_{0} = {(x_{v}) = (t_{v}o_{v}) | (t_{v}) ∈ T_{S}, v ∈ S}. If α is the root of T with respect to P^{+}, | α(t) |= ∏_{v} | α(t_{v}) |_{v} and by abuse of notation α(x) =

∑_{v} α(x_{v}) = ∑ log | α(t_{v}) |_{v} for x_{v} = (t_{v}o_{v}) is a linear weight-function on A_{0} and

H = {x ∈ A_{0} | α(x) = 0} a hyperplane in A_{0}, containing o.

SL_{2}(F) is diagonally embedded in SL_{2}(F_{S}) = ∏_{v} SL_{2}(F_{v}) and by the product-

formula for values we see that T (0_{S}) = {t ∈ T (F)| | α(t) |= 1}, so T (0_{S}) is by Dirichlet’s unit-theorem the product of a finite group and a free abelian group of rank (s − 1). The latter acts on H as a lattice of translations and therefore H/T (0_{S}) is compact. The subcomplex X_{0} := Γ · H of X is then also compact mod Γ. Our aim is to project apartments into X_{0}.

Denote by p_{v} resp. p′_{v} the vertices of C and C′ and consider them as directions in A_{0}, given by half-lines, starting at a center z. We choose a series of centers z_{m}(m ∈ ℕ) in the sector with vertex o and base C: α[(z_{m})_{v}] = m for all v.

Let ρ_{m} be the projection of Z_{t} = X ⨿ Y_{t} into X_{0} with center z_{m}. For the restriction to A_{0} ∪ \overline{A}_{0} we obtain ρ_{m}(p_{v}) = ρ_{m}(p′_{v}) =: q_{v,m} with the following coordinates: α[(q_{v,m})_{w}] = m for all w ≠ v and α[(q_{v,m})_{v}] = −(s − 1)m. The vertices q_{v,m} span a (s − 1)−simplex △_{m} in H and ρ_{m}(\overline{A}_{0}) = △_{m}.

Consider now arbitrary apartments uA_{0} ∈ A_{p}, u ∈ U_{S}. It is well known, that U_{S} = ∏_{v} U(F_{v}) is compact mod U(0_{S}) (see [B1], Satz 3): U_{S} = U(0_{S}) · K, K

compact (if the class-number of 0_{S} is 1, one can choose for K the product ∏_{v} 0_{v} of

the valuation rings). We assume that the origin o is fixed by K.

Take u′ ∈ stabz_{m} ∩ U_{S}, u′ = u · k with u ∈ U(0_{S}), k = (k_{v}) ∈ K and suppose log | u |_{v}= α[(z_{m})_{v}] = m, so m > α[q_{v,m})_{v}], which implies that u fixes none of the vertices q_{v,m} — but u(q_{v,m}) ∈ Γ · H = X_{0}.

Let us now describe the fundamental classes [\overline{A}_{0}] and u[\overline{A}_{0}], elements of H_{s−1}(Y_{t}; ℤ). [\overline{A}_{0}] is given by ∑_{w∈W}(−1)^{lw} · wC; the Weyl-group W of SL_{2}(F_{S}) is

(ℤ/2ℤ)^{s} = {w = ∏_{v} w_{v}^{ϵ_{v}} | ϵ_{v} = 0, 1}, where w_{v} is the involution with w_{v}[P^{+}(F_{v})] =

P^{−}(F_{v}), in particular w_{0} = ∏_{v} w_{v} and C′ = w_{0}C.

For u([\overline{A}_{0}]) we get uw_{v}^{ϵ_{v}}p_{v} = p_{v} for ϵ = 0(U stabilizes C) and uw_{v}^{ϵ_{v}}p_{v} = up′_{v} for ϵ = 1, so all chambers uwC have vertices from {p_{v}} ∪ {up′_{v}}, especially uC = C and uw_{0}C = uC′. In geometric terms: the apartments u\overline{A}_{0} are cross-polytopes (for s = 3 octahedrons) and topologically (s − 1)− spheres.

In the last step we retract these cycles into X_{0} by ρ_{m} (observe that ρ_{m} is compatible with the action of Γ): ρ_{m}(p_{v}) = q_{v} = ρ_{m}(p′_{v}), ρ_{m}(up′_{v}) = uq_{v}. Thus ρ_{m}(u[\overline{A}_{0}]) is a cycle c_{m} ∈ Z_{s−1}(X_{0}; ℤ), whose support is a (s − 1)−sphere S_{m}, consisting of 2^{s} (s − 1)−simplices. We should point out, that different simplices are in different apartments of X with boundary in A_{p}. Moreover these apartments contain the retraction-center z_{m} and therefore the cone C_{m} with base S_{m} and vertex z_{m}, which supports a s−chain C̃_{m}, whose boundary is c_{m}. In the topdimension such a chain C̃_{m} is unique and since z_{m} ∉ X_{0}, we conclude that c_{m} defines a non-trivial class in H_{s−1}(X_{0}, ℤ). In short: S_{m} is a spherical hole in X_{0}. On the other hand lim_{m→∞} z_{m} ∈ C, which means that the cones C_{m} grow out of each subcomplex X′ compact mod Γ : H_{s−1}(X; ℤ) is not essentially trivial (using a filtration (X_{m})_{m∈ℕ_{0}} of X with X_{0} and X_{m}/Γ compact). Then Brown’s criterion ([Br3], thm. 2.2) shows, that SL_{2}(0_{S}) is not of type F_{|S|}.

3.3 The general case

Most problems arise already for a fixed place v ∈ S. For the construction of spheres and cycles it seems convenient not to use complete apartments but only the links of two opposite vertices in Y_{v}. These links have good projections into a hyperplane H in an apartment A of X, on which an arithmetic torus acts cocompactly. The proof of the following proposition can be found in [BW], thm. 2.2 and the literature quoted there.

Proposition 5. Let Q be a maximal F −parabolic subgroup of G, containing a 1−dimensional F −split torus T_{1} with Levi-decomposition Q = Z_{G}(T_{1}) ⋉ R_{u}(Q).

a) There exists a maximal F −torus T in Q, such that:

(i) The maximal F −split torus of T is T_{1}; (ii) T contains a maximal F_{v}− split torus T_{v} for all v ∈ S.

b) T_{S} = ∏_{v∈S} T (F_{v}) acts on an apartment A = ∏_{v} A_{v} of dimension d = ∑_{v} d_{v} in

X = ∏_{v} X_{v}.

c) T_{S} is the product of a compact group and a free abelian group of rank d, which acts on A by translations.

d) T (0_{S}) is discrete in T_{S} and is the product of a finite group and a free abelian group of rank ∑_{v} rank_{F_{v}}(T)− rank_{F} T = d − 1 (“generalized Dirichlet-unit-

theorem”).

e) Fixung an origin o in A, we get a hyperplane H ⊂ A on which T (0_{S}) acts cocompactly.

We concentrate now on a fixed place v ∈ S and consider an apartment A_{v} ⊂ X_{v} with boundary \overline{A}_{v} ⊂ Y_{v} (for simplicity we omit the index v for the details).

\overline{A}_{v} is determined by a pair of opposite chambers C and C′, fixed by F_{v}− minimal parabolic subgroups P and P′. The vertices of C and C′ are stabilized by F_{v}− maximal parabolics, containing P resp. P′. We assume that the F −group Q of prop. 5 is one of them and denote it by Q_{0}, fixing p_{0} ∈ C and its opposites by Q′_{0} and p′_{0}.

There exist several minimal parabolic groups P′_{i}, contained in Q′_{0} (their number

depends on the local Weyl-group), fixing chambers C′_{1}, …, C′_{k}, so the union ⋃_{1}^{k} C′_{i}

is the star of p′_{0} in \overline{A}_{v} and the faces of the C′_{i}, which do not contain p′_{0}, establish the link L(p′_{0}). On the opposite side we have the link L(p_{0}).

Remember that the set of all apartments in \overline{A}_{v}, containing the chamber C is given by the elements of \overline{U}(F_{v}), where \overline{U} is the unipotent radical of P. Set U_{0} := R_{u}(Q_{0}), then U_{0}(F_{v}) ≤ \overline{U}(F_{v}), since Q_{0} ≥ P. This is also true for the other minimal parabolics P_{i} ≤ Q_{0} and if some maximal Q_{i} ≥ P_{i} fixes a vertex p_{i} ∈ L(p_{0}), then the unipotent radical U_{i}(F_{v}) of Q_{i}(F_{v}) has a non-trivial intersection with U_{0}(F_{v}). An element u ≠ 1 from this intersection is not contained in U′_{i}(F_{v}), where U′_{i} is the unipotent radical of Q′_{i}, fixing the vertex p′_{i} opposite to p_{i} — but then u moves p′ ∈ L(p′_{o}).

Now we turn the attention to A_{v}, interpreting the vertices of \overline{A}_{v} as directions in A_{v}. T (F_{v}) acts by translations on A_{v}, in particular T_{1}(F_{v}) on lines with ends p_{0} and p′_{0}. Each of the minimal parabolic subgroups P′_{i} determines a so-called dual root-systems R_{i}^{V} of characters on T (F_{v}). Its basis consists of fundamental weights ω′_{ij}(i = 1, …, k; j = 0, …, d_{v} − 1), which describe the action of T (F_{v}) on the unipotent radicals of maximal parabolic subgroups containing P′_{i}. In particular ω′_{0} := ω′_{i0} for all i belongs to Q_{0}.

If we choose an origin o_{v} ∈ A_{v}, then every x ∈ A is given as x = t_{x}o_{v} with t_{x} ∈ T (F_{v}) and we define ω′_{ij}(x) = log | ω′_{ij}(t_{x}) |_{v} as linear functions on A.

Lemma 1. For H from Prop. 5 H ∩ A_{v} is a hyperplane in A_{v}.

a) The vector ω′_{0} is orthogonal to H ∩ A_{v}.

b) All half-lines in A_{v} with direction ω′_{ij} — starting at a vertex on the same side of H ∩ A_{v} as p_{0} — intersect H ∩ A_{v}.

Proof: a) For a sequence x_{m}(m ∈ ℕ) of vertices in A_{v} with lim_{m→∞} x_{m} = p′_{0} we have lim_{m→∞} vol [stab x_{m} ∩ U′_{0}(F_{v})] = ∞. By reduction theory we know that (U′_{0})_{S} = ∏_{v} U′_{0}(F_{v}) = U′_{0}(0_{S})·K with a compact set K (cf 3.2). Therefore the

finite intersections stab x_{m} ∩ U′_{0}(O_{S}) grow also for m → ∞, so the vertices x_{m} cannot be congruent mod Γ to finitely many ones. Since H is compact mod Γ, the direction w′_{0} can not have a component in H ∩ A_{v}.

b) It is well known that the angles between fundamental weights are acute, if R^{V} is irreducible (cf [B3], 2.2).

If we assume that the origin o = (o_{v})_{v} ∈ ΠA_{v} = A lies in H and observe that the linear function ω′_{0} is defined on all A_{v}, s.th. ω′_{0}[(x_{v})_{v}] = ∑_{v} ω′_{0}(x_{v}), we

obtain — using the product formula for values — an explicit description for H : H = {x ∈ A | ω′_{0}(x) = 0}.

Construction of spheres and cycles in the subcomplex X_{0} := Γ · H of X, which is compact mod Γ — just as H. It will be enough to work in a fixed building X_{v} and take spherical joins in the end. In X_{v} we are not interested in full apartments but only in links of opposite vertices.

In the first step we project the link L(p′_{0}) ⊂ Y_{v} into H ∩ A_{v}, using a series of projection-centers z_{m}, lying on the half-line with vertex o_{v} and direction ρ_{o} with lim_{m→∞} z_{m} = p_{0}. Then all half-lines h_{m,p′} with vertex z_{m} and direction p′ ∈ L(p′_{0})

— or with vector ω′_{ij} — intersect H ∩ A_{v} in a vertex q_{m}. This provides a map of L(p′_{0}) onto the link L(q_{0,m}) in H ∩ A_{v}. In order to simplify the situation we shall not consider the whole link, but only a (d_{v} − 1)−simplex ∑_{m} with vertices from L(q_{0,m}), that will be specified in the following lemma.

In a second step we map ∑_{m} by an element u ∈ stab z_{m} ∩U_{0}(O_{S}) into X_{0} ∩A_{v}. Over F_{v} the element u splits up into root-factors, which are needed to define further simplices connecting ∑_{m} and u(∑_{m}).

Thus the lemma will be an exercise on root-systems. The root-system R for T (F_{v}) with respect to the F_{v}−minimal parabolic group P has a generating set {α_{1}, …, α_{r}} of simple roots, which contains the set △ = {α_{1}, …, α_{r_{0}}} of simple roots over F. We specialize the definition of Q_{0}, assuming that T_{1} = ⋂_{i∈△−{α_{1}}}(ker α_{i})^{0} (where ^{0} denotes the connected component of identity:

Q_{0} = Z_{G}(T) ⋉ U_{0}.

Lemma 2. There exist F_{v}− maximal parabolic subgroups Q_{i}(i = 1, …, r; r = rank_{F_{v}}G), which fix p_{i} ∈ L(p_{0}), and opposites Q′_{i}, fixing p′_{i} ∈ L(p′_{0}) and rootfactors u_{j} ∈ U_{0}(0_{S}) ∩ U_{β}(F_{v}) for some positive root β ∈ R_{+}, also for j = 1, …, r, such that u_{j}(p′_{i}) = p′_{i} for i ≠ j and u_{j}(p′_{j}) ≠ p′_{j}.

For the proof we describe the roots, who contribute factors to the unipotent radicals U_{i} of the groups Q_{i}(i = 0, …, r); we use a suitable order of the simple roots (see the remark afterwards).

Let u ∈ U_{0}(F_{v}) be the product u = ∏_{β} u_{β} with u_{β} ∈ U_{β}(F_{v}) for

β ∈ {α_{1}, α_{1} + α_{2}, …, α_{1} + … + α_{r}} — so we admit no factors, where β has a coefficient > 1.

Now take Q_{1} with U_{1} = R_{u}(Q_{1}), s.th. U_{1} has root-factors U_{j} with j ∈ {α_{r}, α_{r} + α_{r−1}, …, α_{r} + α_{r−1} + … + α_{1}}, perhaps additional ones.

Denote the generators of the local Weyl-group W_{v} with s_{i}(i = 1, …, r), s.th. s_{i}(α_{i}) = −α_{i} and define the groups Q_{i} for i > 1 by Q_{2} = s_{r}(Q_{1}), Q_{3} = s_{r−1}(Q_{2}), …, Q_{r} = s_{2}(Q_{r−1}). We indicate briefly the roots for the unipotent radicals U_{i}. U_{2} : {−α_{r}, α_{r−1}, α_{r−1} + α_{r−2}, …}, U_{3} : {−α_{r} − α_{r−1}, −α_{r−1}, α_{r−2}, …}, …, and finally U_{r} : {−α_{r} − … − α_{r−2}, … − α_{2}, α_{1}}.

Remark: The numbering of roots is not always the usual one. α_{r} must be a long root and for E_{6}, E_{7}, E_{8} the “extra” vertice must be α_{4}.

The result is as follows: Each factor u_{β} from the product u = ∏_{β} u_{β} is contained

in exactly one of the unipotent radicals U_{i}(for i = 1, …, v). If we rewrite u =

∏_{j=1}^{v} u_{j}, we can assume that u_{j} ∈ U_{j}(F_{v}), but u_{j} ∉ U_{i}(F_{v}) for all i ≠ j. Now

u_{j} ∉ U_{i}(F_{v}) implies that u_{j} ∈ U′_{i}(F_{v}) ⊂ Q′_{i}(F_{v}) for the opposite group, which means u_{j}(p′_{i}) = p′_{i} for i ≠ j and since u_{j} ∉ U′_{j}(F_{v}) also u_{j}(p′_{j}) ≠ p′_{j}.

It remains to guarantee, that the factors u_{j} = u_{β} can be chosen in U_{0}(0_{S}). For this purpose we have to make precise the positions of o_{v} and (z_{m})_{v}. We know that U_{0}(F_{v}) = U(0_{S}) · K_{v} with K_{v} compact, so we can suppose that K_{v} fixes o_{v}. For vertices x_{v} ∈ T_{1}(F_{v}) · o_{v} we have β(x_{v}) = α_{1}(x_{v}) for all β ∈ {α_{1}, α_{1} + α_{2}, …, α_{1} + … + α_{r}}, especially we get β(o_{v}) = 0 and we define (z_{m})_{v} by α_{1}[(z_{m})_{v}] = m = β[(z_{m})_{v}]. An arbitrary element u_{β} ∈ U_{β}(F_{v}) can be written in the form u_{β} = u′_{β} ·k_{β} with u′_{β} ∈ U_{0}(O_{S}) and k_{β} ∈ K_{v} with β(u′_{β}) = β(u_{β}) and we have also u′_{β}(p′_{i}) = p′_{i} for u_{β} = u_{j}, i ≠ j and u′_{β}(p′_{j}) ≠ p_{j} — but not u′_{β}(p′_{j}) = u_{β}(p_{j}) in general.

The result can be transmitted to the image of projections into X_{0} = Γ · H.

Corollary 1. For u = ∏_{j=1}^{r} u_{j} ∈ U_{0}(0_{S})∩∏_{β} U_{β}(F_{v})∩stab z_{m} and q_{i} = h_{m,p′_{i}} ∩H ∩A_{v}

there is u(q_{i}) = u_{i}(q_{i}) ∈ X_{0} ∩ A_{v}, since u_{j}(q_{i}) = q_{i} for i ≠ j.

Remark: A simple, but instructive example for this proof is Γ = SL_{4}(0_{S}), s = 1 with r = 3, where q_{1}, q_{2}, q_{3} are vertices of a triangle in L(q_{0}), cutting off the three other vertices of L(q_{0}).

Now it is easy to construct spherical complexes in X_{0} ∩ A_{v}, following a similar pattern as in the example 3.2. We start with the (d_{v} − 1)−simplex ∑_{m} ⊆ L(q_{0}) ⊂ H ∩A_{v} with vertices q_{1}, …, q_{dv}(d_{v} = r), which come from the projection of L(p′_{0}) ⊂ Y_{v} towards the center (z_{m})_{v} ∈ T_{1}(F_{v}) · o_{v}. Now we use a product u =

∏_{j=1}^{r} u_{j} ∈ U_{0}(0_{S}) ∩ stab z_{m} from the corrollary and map ∑_{m} by partial products:

Set J = {1, …, r} = J′ ∪^{·} J′′ and u_{J′} := ∏_{j∈J′} u_{j}, thus u_{J′} fixes the vertices q_{j} with

j ∈ J′′ and for j ∈ J′ we get u_{J′}(q_{j}) = u(q_{j}) ∈ u(∑_{m}) ⊆ u(L(q_{0})). We get 2^{d_{v}} (d_{v} − 1)−simplices, which fit together to a topological (d_{v} − 1)−sphere S_{m}^{d_{v}−1}. Its faces U_{J′}(∑_{m}) are contained in different apartments u_{J′}(A_{v}), determined by the pair (C, u_{J′}(C′)) of opposite chambers in Y_{v}. Moreover all these apartments contain the center (z_{m})_{v}, so S_{m}^{d_{v}−1} can be retracted to (z_{m})_{v}. For (z_{m})_{v} we have α_{1}[(z_{m})_{v}] = m > 0, which means (z_{m})_{v} ∉ X_{0}∩A_{v} and implies that S_{m}^{d_{v}−1} cannot be a boundary in X_{0} ∩X_{v} — by uniqueness in the top-dimension for killing homology: We have a spherical hole in X_{0} ∩ X_{V} .

For the combination of all places v ∈ S we simply have to define the spherical joins of all S_{m}^{d_{v}−1} to obtain topological (d−1)−spheres S_{m}^{d−1} in X_{0}, supporting nontrivial elements in H_{d−1}(X_{0}; ℤ). For lim_{m→∞} z_{m} = p_{0} ∈ Y with α_{1}[(z_{m})_{v}] = m → ∞ for all v ∈ S we have therefore shown that H_{d−1}(X_{0}; ℤ) is not essentially trivial.

Theorem 3. A S−arithmetic subgroup of an absolutely almost simple algebraic group G, defined over a function field F with rank_{F} G > 0 and d = ∑_{v∈S} rank_{F_{v}}G is

not of type F P_{d} and so not of type F_{d}.

Remark: The result can be extended to reductive groups, taking the minimum of the sums of local ranks for the simple factors.

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