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

Helmut Behr, Frankfurt am Main

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 at last 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_{∞}.

This paper will give a proof of the positive part, using two old results on cohomology. They have the advantage not to need such precise local informations, which are necessary in (almost) all proofs of the results mentioned above, using 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 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]).

It seems plausible, that the information on the top-cohomology should provide a new proof for being not of type F P_{∞}, but I don’t know how to use it.

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 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) ⊃ C_{c}^{∞}(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′), −∂′_{k−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−arithemtic 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 pathes 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, then 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)_{1} 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 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 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 concrete 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 (of [Sp1], section 19 or [Sp2], 6.9).

Recall that an element of H_{c}^{n}(X; M) with an arbitrary coefficient-module M is then given by an abelian group homomorphism φ on C_{n}(X), that 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 index c means compact support, so it consists in X of finitely many polysimplices. The action of Γ on φ is defined by (φγ)(c) = φ(γc).

Maybe it is more convenient to consider Γ−homomorphisms φ̃ instead of φ, whose support is compact mod Γ, i.e. represented by finitely many polysimplices. If a module of φ′s is finitely generated, then the union of all supports of φ̃’s is also compact mod Γ. Moreover we shall regard two variants of complexes C′, either it is a chain complex of finitely generated free Γ−modules or it is f(C′) = C(X′), where X′ is a subcomplex of X of lower dimension, X′/Γ finite: then f becomes inclusion and ∂′ the restriction of ∂ to X′.

Now we shall continue the induction, beginning with C′_{0} = ℤΓ; f_{0} : C′_{0} → C_{0} with f_{0}(Σz_{γ}γ) = Σz_{j}(γx_{0}), so f_{0} has finite kernel, since stab_{Γ}x_{0} is finite; X′_{0} = Γ · x_{0}. We define the (abelian-group-) homomorphism φ : C′′(k)_{k} = C_{k} × C(k − 1)_{k−1} → C_{k−1} by φ(c, c′) := ∂_{k}(c), assuming that C′_{k−1} is a finitely generated free ℤΓ−module and f_{k−1} has finite kernel.

Since C′(k − 1)_{k} = 0 we have B_{k}(C′′(k)) = B_{k}(C) and φ vanishes on B_{k}(C), so it is an element of Z^{k}(C′′(k), C_{k−1}). We modify φ in its cohomology class by ψ := π_{1} ∘ ∂′′_{k} : C′′(k)_{n} → C_{k−1}, ψ(c, c′) = π_{1}(∂_{k}c − f_{k−1}c′, ∂′_{k−1}c′) = (∂_{k}c − f_{k−1}c′); ψ vanishes on Z^{k}(C′′(k), C_{k−1}) and φ − ψ induces the same element in H^{k}(C′′(k), C_{k−1}) as φ : \overline{φ}. We obtain \overline{φ}(c, c′) = f_{k−1}(c′); its restriction to H_{k}(C′′(k)) is injective in the first component, since H_{k}(C) = 0 and has finite kernel in the second, since f_{k−1} has. Because then ∂′_{k−1}(c′) = 0, \overline{φ} induces a surjective homomorphism with finite kernel from H_{k}(C′′(k)) to f_{k−1}[H_{k−1}(C′(k − 1)] or even onto

f_{k−1}(H_{k}(C′′(k))), since H_{k}(C′′(k)) ≃ H_{k−1}(C′(k−1)), due to sequence (1), given by (c, c′) ↦ c′.

For isotropic groups G X/Γ is not compact, so we have to consider an ascending sequence X_{m}(m ∈ ℕ) of subcomplexes with ⋃_{m} X_{m} = X and X_{m}/Γ compact

(“filtration”). Denote by φ_{m} the restriction of φ to C(X_{m})_{k} × C′(k − 1)_{k−1} and observe by induction, that the support X′_{k−1} of C′(k − 1) is also compact: we may suppose that X′_{k−1} ⊆ X_{m}. Thus we have φ_{m} ∈ H_{c}^{k}(C′′(k); C_{k−1}).

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 an arbitrary ℤΓ− module M.

According to Borel-Serre (see thm.1) we know, that H_{c}^{n}(C; M) = 0 for n ≤ d − 1, which implies H_{c}^{k}(C′′(k); M) ≃ H_{c}^{k−1}(C′(k); M) for k ≤ d − 1. The cohomology of the complex C′(k) of finitely generated free ℤΓ−modules is also finitely generated and so is H_{c}^{k}(C′′(k); M).

We apply this result to the homomorphisms \overline{φ}_{m} with M = C_{k−1} : All \overline{φ}_{m} are ℤΓ−linear combinations of finitely many elements, say ϕ_{1}, …, ϕ_{s} ∈ H_{c}^{k}(C′′(k); M) the union of the supports of all ϕ_{i} is contained in some X_{m_{0}}(m_{0} ∈ ℕ), which means \overline{φ} = \overline{φ}_{m}V m ≥ m_{0} and \overline{φ} = 0 outside X_{m_{0}}. Consider now finitely generated submodules of H_{k}(C′′(k)) : each of them is mapped by some \overline{φ}_{m} to f_{k−1}(H_{k−1}(C′(k)))

with finite kernel. Since lim_{\overrightarrow{m}} φ_{m} = φ, we conclude that H_{k}(C′′(k)) is also finitely

generated; take z_{i} = (c_{i}, c′_{i}) ∈ Z_{k}(C′′(k)), o = 1, …, r_{k} as generators and define C′_{k} as the free module (ℤΓ)^{r_{k}}, projecting to ⨁_{1}^{r_{k}} ℤΓ · z_{i} and ∂′_{k} and f_{k} as in 2.1.

Since the stabilisators of the c_{i} are finite, f_{k} : C′_{k} → C_{k} has finite kernel and for the complex C′(k) with C′(k)_{k} = C′(k) we get H_{k−1}(C′(k)) = 0. Further-

more X′_{k} = ⋃_{1}^{r_{k}} supp(C_{i}) ∪ X′_{k−1} is a subcomplex of X with X′_{k}/Γ compact and

H_{k−1}(C(X′_{k})) = 0.

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}^{0}(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.

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 Lectures Notes 1641 (1996).

[B1] H. Behr: Endliche Erzeugbarkeit arithmetischer Gruppen über Funktionenkörpern; Inv. Math. 7 (1969), 1-32.

[B2] H. Behr: SL_{3}(𝔽_{q}[t]) is not finitely presentable; Proc. Symp. “Homological group theory (Durham 1977)”; London Math. Soc. Lect. Notes Ser. 36, 213-224.

[B3] H. Behr: Arithmetic groups over function fields I. A complete characterization of finitely generated and finitely presented arithmetic subgroups of reductive algebraic groups; 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, Combinatorial, Arithmetic Aspects. London Math. Soc. Lect. Notes Series 311 (2004), 27-42.

[BS] A. Borel, J.P. Serre: Cohomologie d’immeubles et de groupes S-arithmétiques, Topology 15 (1976), 211-223.

[BT] A. Borel, J. Tits: Groupes réductifs; Publ. Math. I.H.E.S. 27 (1965), 55-150.

[Br1] K. Brown: Homological criteria for finiteness; Comm. Math. Helv. 50 (1975), 129-135.

[Br2] K. Brown: Cohomology of groups; Springer GTM 87 (1982).

[Br3] K. Brown: Finiteness properties of groups; Journal of Pure and Applied Algebra 44(1987), 45-75.

[BW] K.U. Bux, K. Wortman: Finiteness properties of arithmetic groups over function fields; Inv. Math. 167 (2007), 355-378.

[H] G. Harder: Minkowskische Reduktionstheorie über Funktionenkörpern; Inv. Math. 7 (1969), 33-54.

[S] J.P. Serre: Arbres, amalgames, SL_{2}; astérisque 46 (1977), (≃ Trees; Springer 1980)

[Sp1] E.H. Spanier: Cohomology theory for general spaces; Ann. of Math. 49(1948), 407-427.

[Sp2] E.H. Spanier: Algebraic topology; McGraw-Hill (1966).

[St1] U. Stuhler: Zur Frage der endlichen Präsentierbarkeit gewisser arithmetischer Gruppen im Funktionenkörperfall; Math. Ann. 224 (1976), 217-232.

[St2] U. Stuhler: Homological properties of certain arithmetic groups in the function field case; Inv. Math. 57 (1980), 263-281.