Higher finiteness properties of S-arithmetic groups in the function field case I

by Helmut Behr (Frankfurt a.M.)

It is well known that S-arithmetic subgroups of reductive algebraic groups over number fields have “all” finiteness properties (see [BS 2]). On the contrary there exist many counterexamples in the function field case. Let F be a finite extension of 𝔽_{q}(t), G an almost simple algebraic group of F -rank r, 0_{S} an S-arithmetic subring of F with # S = s, r_{v} the F_{v}-rank of G over the completion F_{v} of F for v ∈ S, and finally Γ a S-arithmetic subgroup of G(F).

We are interested in the following question: Is it true that

Γ is of type F_{n−1} but not F_{n} iff r > 0 and ∑_{v∈S} r_{v} = n ?

(For the definition of finiteness properties, cf the introduction of [Ab].) The answer is yes in the following cases:

(a) G = SL_{2}: see [St 2].

(b) n = 1 or 2 (finite generation or finite presentability): see [B 2].

(c) G classical, 0_{S} = 𝔽_{q}[t] under the assumption that q is big enough compared with r: see [Ab] and [A] for SL_{n}.

In particular it is not known if the assumption in (c) is necessary. For r = 0, in the so-called cocompact case, Γ is of type F_{∞} (cf. [BS 2]).

This paper is an attempt to attack this question with some new methods — old in other contexts. First of all, inspired by the work of Serre, Quillen, Stuhler and Grayson (cf. [G1,2]), we use semi-stability for reduction theory, and the idea to determine the homotopy type of the boundary of the unstable region by retraction.

In this part we only deal with Chevalley groups G and arithmetic rings O_{S} for # S = 1. The groups G(F) and Γ act on the Bruhat–Tits building X = X_{v}, corresponding to G and F_{v}; Γ leaves the unstable region X′ invariant. X′ has a cover whose nerve is given by the spherical Tits building X_{0}, so it is (r − 1)- spherical. The retraction to its boundary is not possible as in the number field case, since the geodesic lines are branching (discretely).

Therefore we have to “split up” X′ into apartments, thereby constructing a bigger complex X̃′, which has a cover with nerve OppX_{0}, defined by an opposition relation in X_{0}. This complex was first considered by Charney for G = GL_{n}, by Lehrer and Rylands for classical groups who called it “split building”, finally v. Heydebreck showed in the general case that this complex is also (r − 1)-spherical — so is X̃′.

X̃′ can be retracted to its boundary Ỹ, but Ỹ is not finite modΓ. Thus we have to consider a subcomplex X̃′_{Γ}, where opposition is defined only with respect to Γ and to show that X̃′_{Γ} is a deformation retract of X̃′. Now we obtain that Ỹ_{Γ} is finite modulo Γ and can deduce the F_{n−1}-property of Γ. For the negative part, i.e. Γ is not of type F_{n}, one should come back to filtrations, the method used for the proofs of (a), (b) and (c) above, but for the moment I have no detailed argument. Thus we sketch the proof of the following

Theorem. The S-arithmetic subgroup Γ = G(O_{S}) of a simply connected almost simple Chevalley group G of rank r is for s = 1 of type F_{r−1}. Conjecture: Γ is not of type F_{r}.

I hope that this program will turn out to be useful even in more general situations: For coefficient rings which are defined by more than one prime or, on the other side, for non-split groups.

I am grateful to Peter Abramenko for constructing a very instructive counterexample to an earlier version of this paper, and I would like to thank him, Kai-Uwe Bux and Anja von Heydebreck for helpful discussions, and Mrs. Christa Belz for carefully writing several versions.

1. Notations

Let us denote by F a finite extension of the field of rational functions 𝔽_{q}(t) in t

with coefficients in the finite field 𝔽_{q}, q = p^{m};

F̂ = F_{v}  the completion of F with respect to the valuation v of F;
O and Ô  the valuation rings with respect to v in F or F̂;

G a simply connected almost simple Chevalley group, defined over F;

r  the F -rank of G, I = {1, …, r};
T  a maximal (split) F -torus of G;
Δ = {α_{i}}_{i∈I} a set of simple roots of G with respect to T;

P_{Δ_{0}} a parabolic subgroup of G of cotype Δ_{0}, Δ_{0} ⊆ Δ, which means that Δ−Δ_{0} is a set of simple roots for the semi-simple part of P_{Δ_{0}}, especially

B = P_{Δ}  the Borel subgroup, defined by Δ, and
P_{α}  the maximal parabolic subgroup for Δ_{0} = {α}.

X the Bruhat–Tits–building, corresponding to G and v with its simplicial structure and its metric topology;

A = X_{T}  the apartment of X corresponding to T, thus A ∼ ℝ^{r};

{α_{i}(x)}_{i∈I} the coordinates of x ∈ A which means by abuse of notation the following: If x = t · x_{0}, x_{0} the “origin” of A, t ∈ T (F̂),

then α_{i}(x) := −v(α_{i}(t));
X_{0}  the spherical Tits building of G(F);
Γ  the S-arithmetic subgroup of G(F) for S = {v}.

2. Reduction Theory and the Unstable Region

We shall use reduction theory for arithmetic groups over function fields in the version described by Harder in [H2], 1.4. He defines

π(x, P) := vol (K_{x} ∩ U(F̂))

for a special point x ∈ X, corresponding to a maximal compact subgroup K_{x} of G(F̂) and a F -parabolic group P and its unipotent radical U; the volume vol comes from the adelic Tamagawa measure. The function

d_{P} (x) := log_{q} π(x, P)

can be extended by linear interpolation to all points x in an apartment A = X_{T}, defined by a maximal split F̂-torus T, contained in P and thereby uniquely for all x ∈ X. We may consider d_{P} as a co-distance with respect to the simplex σ_{P}, given by P in the spherical building X_{∞} at infinity (cf. [Br2], VI.9). For the action of T (F̂) on X_{T} via ad T we have the formula

d_{P} (t · x) = d_{P} (x) + log_{q} |δ_{P} (t)|

where δ_{P} is the character “sum of roots in U”, which is a multiple of the dominant weight ω_{P} and the q-logarithm is the negative additive valuation −v(δ_{P} (t)). For each Borel group B over F and its set Δ = {α_{1}, …, α_{r}} of simple roots (with respect to a F -torus T), the maximal parabolic groups P_{α} (α ∈ Δ) containing B and their fundamental weights ω_{P_{α}}, one has

α = ∑_{β∈Δ} c_{α,β}ω_{P_{α}} = ∑_{β∈Δ} c′_{α,β}δ_{P_{α}}

where c_{α,β} are the integral coefficients of the Cartan-matrix, such that c′_{α,β} ∈ ℚ; in particular, c′_{α,α} is positive and c′_{α,β} for β ≠ α is zero or negative (for at most 3 β’s). Using these coefficients, Harder defines numerical invariants

n_{α}(x, B) := ∏_{β∈Δ} π(x, P_{β}).

Again we pass to the additive version, setting

c_{B,α}(x) := log_{q} [n_{α}(x, B)]

and obtain for each b ∈ B(F) the relation

c_{B,α} (b · x) = c_{B,α}(x) + log_{q} |α(b)|
= c_{B,α}(x) − v(α(b))

c_{B,α} is an affine linear function on the apartment X_{T}; we define the origin O_{B} by c_{B,α}(O_{B}) = 0 for all α ∈ Δ and by abuse of notation α(t · O_{B}) := −v(α(t)) for t ∈ T (F), thus we get by linear interpolation a set of affine coordinates {α_{1}(x), …, α_{r}(x)} for each point x ∈ X_{T}.

Now we are able to state the main theorems of reduction theory (for Chevalley groups):

(A) There exists a constant C_{1} such that for all x ∈ X there is a F -Borel group B with c_{B,α}(x) ≥ C_{1} for all α ∈ Δ; then x is called “reduced with respect to B”.

(B) There exists a constant C_{2} ≥ C_{1}, such that for x ∈ X reduced with respect to B and B′, and c_{B_{α}}(x) ≥ C_{2} for all α ∈ Δ_{0} ⊆ Δ, P = P_{Δ_{0}} ⊇ B, it follows P ⊇ B′; then x is called “close to P ”, P is uniquely determined.

(C) There exists a constant C_{3} ≥ C_{2}, depending on the arithmetic group Γ, such that for x ∈ X, reduced with respect to B and with c_{B,α}(x) ≥ C_{3} for all α ∈ Δ_{0} ⊆ Δ, we have for the unipotent radical U of the parabolic group P = P_{Δ_{0}} ⊇ B

U(F̂) = (U(F̂) ∩ K_{x})(U(F) ∩ Γ);

x is then called “very close to P ”.

(D) For each constant C ≥ C_{1} the set

X_{C} := \{x ∈ X | \begin{matrix}c_{B,α}(x) ≤ C for all α ∈ Δ and all B \\ for which x is reduced with respect to B \end{matrix} \}

is Γ-invariant and X_{C}/Γ is compact.

(E) The number of Borel subgroups over F of G belongs to finitely many classes under Γ-conjugation (see [B1], 8).

Remark. The constant C_{1} can be chosen as C_{1} ≤ −2g − 2(h − 1) where g denotes the genus of F and h is a “class-number” (for the precise definition see [H1], 2.2.6). For example, if Γ = SL_{n} (𝔽_{q}[t]) we may use C_{1} = 0, but in general C_{1} is negative.

We define the cone or sector of points in X_{T}, reduced with respect to B ⊃ T by

D_{B,T} := {x ∈ X_{T} | α_{i}(x) ≥ C_{1} for i = 1, …, r}

Warning: For different Borel groups B and B′, containing the same torus T, the origins O_{B} und O_{B′} must not coincide and therefore the sectors D_{B,T} , B ⊃ T do not cover in general the apartment X_{T}: see example below.

For a F parabolic group P of cotype Δ_{0} ≠ ∅, we denote by X′_{P} the set of all points x ∈ X which are close to P:

X′_{P} := \{x ∈ X | \begin{matrix}c_{B,α}(x) ≥ C_{1} for all α ∈ Δ \ Δ_{0} \\ c_{B,α}(x) ≥ C_{2} for all α ∈ Δ_{0} \end{matrix} for all B ⊆ P \}

or X′_{P} := ⋃_{B⊆P} D_{B} := ⋃_{B⊆P} ( ⋃_{T ⊆B} D_{B,T} ∩ X′_{P} )

and call

X′ := ⋃_{P} X′_{P} = ⋃_{P max} X′_{P}

the unstable region of X; the name is given in analogy to the description with vector bundles for the group G = SL_{n} (cf. [G1], 4).

For a F -parabolic group Q let P run over all maximal F -parabolic groups which contain Q; then we have

X′_{Q} = ⋂_{P ⊇Q} X′_{P} .

We obtain a polyhedral decomposition of X′, defining

X′′_{Q} := \overline{X′_{Q} \ ⋃_{Q_{1}⫋Q}(X′_{Q} ∩ X′_{Q_{1}})} .

In the special case, where C_{1} = 0, we have in a fixed sector D_{B,T} the following descriptions:

X′_{Q} ∩ D_{B,T} = {x ∈ D_{B,T} | α(x) ≥ 0 for all α ∈ Δ, α(x) ≥ C_{2} for all α ∈ Δ_{0}}
X′′_{Q} ∩ D_{B,T} = {x ∈ D_{B,T} | 0 ≤ α(x) ≤ C_{2} for all α ∈ Δ \ Δ_{0},
α(x) ≥ C_{2} for all α ∈ Δ_{0}}

where Q = P_{Δ_{0}}. In particular for Q = B, which means Δ_{0} = Δ, X′′_{B} ∩ D_{B,T} is a cone inside D_{B,T}, for Q = P maximal, i.e. Δ_{0} = {α}, we get for X′′_{P} ∩ D_{B,T} a cylindric convex set, furthermore infinite prisms etc.

Finally we have X′_{P} = ⋃_{Q⊆P} X′′_{Q}.

Remark. Assume we have an enumeration of the set of simple roots, given by a type function on the vertices of the spherical building X_{0}, then for x ∈ X′′_{Q} the set of maximal parabolic subgroups P containing Q defines a chain which generalizes the “canonical filtration” of vector bundles for G = SL_{n} (cf. [G1]) or respectively lattices in the number field case (cf. [St1] and [G2]).

Above all we are interested in the boundary Y := ∂X′ of the unstable region, which can be described for a parabolic group Q of cotype Δ_{0} ≠ ∅ as follows:

Y_{Q} := ∂X′′_{Q} := \{x ∈ X′′_{Q} | \begin{matrix}c_{β,α}(x) ≥ C_{1} & for all α ∈ Δ \ Δ_{0} and all \\ & B ⊆ Q \\ c_{β,α}(x) ≥ C_{2} & for all α ∈ Δ_{0} and equality for \\ & at least one B ⊆ Q \end{matrix} \}

Y = ∂X′ := ⋃_{Q} ∂X′′_{Q} .

In the next step we distinguish geodesic lines in X′′_{Q}: A point x ∈ X′′_{Q} with coordinates α(x) for an appropriate B determines uniquely a boundary point y ∈ Y_{Q} by setting α(y) = α(x) for all α ∈ Δ − Δ_{0} and α(y) = C_{2} for all α ∈ Δ_{0}, the segment \overline{xy} lies on a geodesic. The “geodesic action” on this line in the apartment X_{T} is given by the torus T_{Δ_{0}} := {t ∈ T | α(t) = 0 for all α ∈ Δ − Δ_{0}}, contained in the radical of Q = P_{Δ_{0}}, centralizing its semi-simple part. Along these geodesic lines we can define a retraction of X′′_{Q} to its boundary Y_{Q}, for instance parametrized by the distance function d_{Q}. Therefore the local definitions fit together for X′_{Q}, but unfortunately they define no retraction from X′_{Q} to ∂X′_{Q} since the geodesic lines are branching into different apartments.

We shall need a further retraction from the sets X′_{P} to “infinity” along geodesics of “type P_{Δ_{0}}”, given by the action of T_{Δ_{0}}, see next section.

Example. G = SL_{n}, Γ = SL_{n}(𝔽_{q}[t])

1. In this case Γ admits a strict simplicial fundamental domain D which is a sector D_{B,T} for a fixed pair T ⊂ B: see [Ab], I.3); this result can also be deduced from reduction theory with Siegel sets. This corresponds to the fact that we can choose C_{1} = 0, C_{2} = 1 in Harder’s theory for this case. One may then define the polyhedral decomposition locally in D and extend it to X by the action of Γ.

2. In order to show that origins O_{B} and O_{B′} of different sectors in an apartment must not coincide, we use n = 3: Denote by B^{+} the upper triangular

matrices in SL_{3}, by B^{−} = w B^{+}w^{−1} with w = ( \begin{matrix}0 & 0 & 1 \\ 0 & −1 & 0 \\ 1 & 0 & 0 \end{matrix}) the lower

triangular matrices, define B′ = g · B^{−} := g B^{−}g^{−1} = g w B^{+}w^{−1}g^{−1} with

g = ( \begin{matrix}1 & 0 & t^{−n} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ), n ∈ ℕ, such that B^{+} and B′ are opposite Borel groups, defining an apartment A. We obtain an equation gw = γwb with γ ∈ Γ, b ∈ B (𝔽_{q}(t)), explicitly

( \begin{matrix}t^{−n} & 0 & 1 \\ 0 & −1 & 0 \\ 1 & 0 & −1 \end{matrix}) = ( \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ t^{n} & 0 & −1 \end{matrix})( \begin{matrix}t^{−n} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & t^{n} \end{matrix} )( \begin{matrix}1 & 0 & t^{−n} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ).

Compute

c_{B′,α′}(O_{B}) = c_{gw·B^{+}, gw(α)}(gw(O_{B}))

(since w(O_{B}) = O_{B} and g fixes a half-plane containg O_{B})

= c_{γwB^{+},γw(α)}(γw · b(O_{B}))
= c_{B^{+},α}(b(O_{B}))

(by left-invariance of the measure)

= c_{B^{+},α}(O_{B}) + v(α(b)) = 0 − n

which is valid for α = α_{1} and α = α_{2}, thus O_{B} ≠ O_{B′}: to get O_{B′}, we have to shift O_{B} in “direction of B′”, precisely: with the coordinates α_{1}, α_{2} corresponding to B one has O_{B′} = (−n, −n).

3. Compactification of the Bruhat–Tits Building

For the boundary at infinity of X we do not use the topologization of the building at infinity due to Borel–Serre; it is more convenient to have the compactification, constructed by Landvogt in [L], but we restrict it to the part defined over F.

For a local field F̂ and a reductive algebraic group H denote by X(H) the Bruhat– Tits building for the pair (H, F̂), then define

\overline{X} := \overline{X}(G) := ⋃_{P ∈P} X (P/R_{u}(P)) ,

where P is the set of all parabolic F -subgroups of G and R_{u}(P) the unipotent radical of P (cf. [L], 14.21). \overline{X} is equipped with a topology which comes from the F̂-analytic topology on G(F̂) and the compactification of apartments, described below, and it induces the metric topology on each of the buildings X(P/R_{u}(P)). Consequently we consider only the — incomplete, but good (cf. [Br2], VI.9) — apartment system A, defined over F, which is in 1-1-correspondence with the apartment system A_{0} of the Tits building X_{0} of G(F).

For A ∈ A denote by V the underlying F̂-vectorspace, by Σ the Coxeter complex with respect to G in V, by C a chamber of Σ and by Δ(C) a set of simple roots, such that C = {x ∈ A | α(x) ≥ 0 for all α ∈ Δ(c)}. For an open face C′ of C, set Δ(C′) := {α ∈ Δ(C) | α_{|C′} > 0} and denote by ⟨C′⟩ the subspace of V, generated by C′. V^{C} := ⋃_{{C′∈Σ \atop C′⊆\overline{C}}} V/⟨C′⟩ is called the corner defined by C.

Provide ℝ̃ := ℝ ∪ {∞} with its natural topology and topologize V^{C} in such a way that the map f : V^{C} ⟶ ℝ̃^{n}, given by

f(x + ⟨C′⟩) := \{ \begin{matrix}∞ & for α ∈ Δ(C′) \\ α(x) & for α ∈ Δ(C) \ Δ(C′) \end{matrix}

is a homeomorphism.

A set U ⊆ \overline{V} := ⋃_{C′∈Σ} V/⟨C′⟩ is called open if U ∩ V^{C} is open for all chambers

C ∈ Σ; by that \overline{V} becomes compact and is called the compactification of V. \overline{A} := A ×^{V} V^{Σ} is then the compactification of A with corners A^{C} (cf. [L], §2).

We abbreviate in the following: X(P) := X(P/R_{u}(P)), and we define the boundary of \overline{X} by

∂\overline{X} := \overline{X} \ X = ⋃_{P ≠G} X(P) .

The closure of X(P) in \overline{X} is given by ⋃_{Q⊆P} X(Q); we shall also need

X_{P} := X ∪ X(P).

Our next aim is to determine the homotopy type of the unstable region X′, using the cover with the sets X′_{P} , P a maximal parabolic F -group. The nerve of this cover is the spherical Tits building X_{0} which is known to be (r − 1)-spherical. For this purpose we have to show that the sets X′_{P} and their intersections X′_{Q} (Q an arbitrary F -parpabolic group) are contractible, and to prove this we construct retractions to infinity, more precisely to X(Q), defined by the geodesic action of the torus T_{Δ_{0}} for Q = P_{Δ_{0}}. To describe it in a sector D_{B,T} , T ⊇ T_{Δ_{0}}, it is helpful not to use all local coordinates α for D_{B,T} (α ∈ Δ), but only those α, lying in Δ−Δ_{0} and to complete them with the functions d_{P} for all P = P_{α}, α ∈ Δ_{0} (this is admissible since the roots in Δ − Δ_{0} and the fundamental weights for Δ_{0} are linearly independent). Then we can define the map

r_{Q,B,T} : D_{Q,B,T} × [0, ∞] ⟶ D_{Q,B,T} (Q ⊇ B)

for D_{Q,B,T} := \overline{D_{B,T} ∩ X′_{Q}} ∩ X_{Q} where the closure is meant in \overline{X}, given by r_{Q,B,T} (x, t) = x_{t} with

α(x_{t}) = α(x)  for all α ∈ Δ − Δ_{0} and x ∈ X
d_{P} (x_{t}) = d_{P} (x) + t for all P = P_{α}, α ∈ Δ_{0} and x ∈ X
α(x) = x  for all α ∈ Δ, x ∈ \overline{X} \ X.

For different tori T and T′, containing T_{Δ_{0}}, points x ∈ D_{B,T} and x′ ∈ D_{B′,T′} can have the same image for t = ∞ in X(Q), described by different systems of coordinates α, coming from the apartments X_{T} and X_{T′} respectively, but the coordinates d_{P} for P ⊇ Q are defined independently from these apartments. Thus the maps r_{Q,B,T} fit together, defining for t = ∞ a retraction

r_{Q} : \overline{X′_{Q}} ∩ X_{Q} ⟶ X(Q) .

The map r_{Q} is continuous since its restrictions to the sectors D_{B,T} are fibrations. Moreover, the map r_{Q} is surjective: For each point x ∈ X(Q) we find a point x′ projecting to x for sufficiently large values d_{P} (x′) for all P ⊇ Q such that x′ is close to Q, and therefore exists B ⊆ Q for which x′ is reduced, so x′ ∈ D_{B,T} for some T ⊆ B and x′ ∈ D_{B,T} ∩ X′_{Q}. Finally the affine building X(Q) is contractible, thus by the retraction r_{Q} the set \overline{X′_{Q}} ∩ X_{Q} is also contractible and as a metrizable manifold the same is true for its interior X′_{Q} (cf. [BS1], 8.3.1).

Proposition 1. The unstable region X′ is (r − 1)-spherical.

Proof: X′ = ⋃_{P ∈P_{max}} X′_{P} with P_{max} := {P maximal F -parabolic in G}, the non-

empty intersections of the covering sets X′_{P} are of type X′_{Q}, Q F -parabolic, and we have seen above that alle this sets are contractible. The covering sets are closed and the cover is locally finite because X is a locally finite simplicial complex. Its nerve is given by the spherical Tits building X_{0} as an abstract complex which is known to be (r − 1)-spherical. Thus we obtain that X′ is (r − 1)-spherical, using the same theorem as Borel–Serre in [BS1], 8.2.

Remark. For the group G = SL_{n} (or G = GL_{n}) proposition 1 was proved by Grayson with a similar argument using vector bundles (cf. [G1], thm. 4.1). The same idea can be used for ∂\overline{X} := \overline{X} − X = ⋃_{P ≠G} X(P). We have the natural

cover ∂\overline{X} = ⋃_{P ≠G} \overline{X(P)} with \overline{X(P)} = ⋃_{Q⊆P} X(Q); all these sets are contractible as

Bruhat–Tits buildings or closures of them and their intersection pattern is given again by X_{0}. So we get the

Corollary. ∂\overline{X} is (r − 1)-spherical.

4. Buildings with Opposition

(a) In each apartment A_{0} of a spherical building X_{0} there exists a natural opposition involution. If A_{0} is described as an abstract Coxeter complex Σ = Σ(W, S) with group W and generating set S, W_{J} = ⟨J⟩ for J ⊆ S, i.e. Σ = {wW_{J} | w ∈ W, J ⊆ S} and w_{0} denotes the element of maximal length in W, then define

op_{Σ}(w W_{J}) := ww_{0}W_{w_{o}Jw_{0}} ;

expecially if the Coxeter diagram has no non-trivial symmetry, then w_{0}Jw_{0} = J for all J.

If X_{0} is the spherical Tits building of a group G(F) (G reductive, F a field), the simplices of X_{0} may be identified with the proper F -parabolic subgroups of G(F). Each such group has a Levi decomposition P = L ⋉ R_{u}(P), and two parabolics are called opposite if they have a common Levi subgroup, more precisely,

P op P′ : ⟺ P ∩ P′ is a Levi subgroup of P and P′.

[R_{u}(P)](F) acts simply transitive on the set of all parabolic subgroups opposite P (cf. [BT], § 4), thus we can identify them with the elements of this radical if we distinguish one opposite group.

(b) Pairs of opposite simplices of a spherical building with incidence in both components provide again a simplicial complex. It was introduced by R. Charney (see [C]) for G = GL_{n}, even over Dedekind domains in the language of flags; she showed that it has the same homotopy type as the spherical building of GL_{n} itsself. Lehrer and Rylands (see [LR]) defined such a complex for reductive groups G — they called it the “split building” of G — and proved the corresponding homological result for types A_{n} and C_{n}. A. von Heydebreck (see [vH]) considered this complex for arbitrary spherical buildings and showed that it is also (n−1)-spherical in dimension n. We use the definition

Opp X_{0} := {(P, P′) | P op P′}.

(c) Moreover, we need a subcomplex of OppX_{0}, where the opposition relation is defined with respect to Γ.

As a first step we distinguish an apartment A_{1} = X_{T_{1}} of X, T_{1} a maximal split F -torus such that N(T_{1}) ∩ Γ contains (a copy of) the Weyl group W of X_{0} (for instance, A_{1} could contain a vertex with stabilizer G(Ô) ⊃ G(𝔽_{q}) ⊃ W). We fix a Borel group B_{1} ⊃ T_{1} and its opposite B′_{1} in A_{1}. The choice of B′_{1} defines an identification of Opp B_{1} := {B′ | B′ op B_{1}} with U_{B_{1}}(F), and we can consider the subset Opp_{Γ} B_{1}, corresponding to U_{B_{1}}(F) ∩ Γ =: U_{1} ∩ Γ such that

Opp_{Γ} B_{1} := {B′ = γ_{1}B′_{1}γ_{1}^{−1} | γ_{1} ∈ U_{1} ∩ Γ}.

We extend this notion Γ-invariant: For B = γB_{1}γ^{−1} with γ ∈ Γ, the element γ is determined up to B_{1}(F) ∩ Γ, so we obtain different opposite Borel groups B′ = δB′_{1}δ^{−1} with δ ∈ γ · (U_{1} ∩ Γ) — neglecting the torus component in T_{1} ⊂ B_{1} since it fixes also B′_{1}. Consequently the identification of Opp B with U_{B}(F) depends on the choice of δ, but this has no influence on the definition

Opp_{Γ}B := {B′ = γ′B′(γ′)^{−1} | γ′ ∈ U_{B}(F) ∩ Γ}

because U_{B} = γU_{1}γ^{−1}, which implies with u, u′ ∈ U_{1} ∩ Γ:

γ′B′(γ′)^{−1} = γu′γ^{−1}γuB′_{1}(γu)^{−1}(γu′γ^{−1})^{−1} = γu′uB′_{1}(γu′u)^{−1}

thus Opp_{Γ}B = γ · Opp_{Γ}B_{1}γ^{−1}.

In general, not all F -Borel groups are conjugate under Γ; there exist finitely many Γ-conjugacy classes (see part E of reduction theory). We fix a set B_{1}, B_{2} = g_{2}B_{1}g_{2}^{−1}, …, B_{h} = g_{h}B_{1}g_{h}^{−1} (g_{i} ∈ G(F)) of repesentatives and also of their opposite groups B′_{1}, B′_{2} = g_{2}B′_{1}g_{2}^{−1}, …, B_{h} = g_{h}B′_{1}g_{h}^{−1}, and define in the same way as above

Opp_{Γ}B_{i} := {B′ = γ_{i}B′_{i}γ_{i}^{−1} | γ_{i} ∈ U_{B_{i}}(F) ∩ Γ}, i = 1, …, h

and for B = γB_{i}γ^{−1}, B′ = γB′_{i}γ^{−1}

Opp_{Γ}B := {B′ = γ′B(γ′)^{−1} | γ′ ∈ U_{B}(F) ∩ Γ}

which does not depend on the special choice of B′ (but we don’t have g_{i} Opp_{Γ}B_{1}g_{i}^{−1} = Opp_{Γ}B_{i} in general).

Finally we can make the same procedure with parabolic groups, starting with the set of standard parabolic groups Q_{1} containing B_{1} and their oposites Q′_{1} ⊇ B′_{1}. Since Q_{1} and Q′_{1} have a Levi subgroup in common, we obtain all Γ-opposites of Q_{1} by conjugation of Q′_{1} with elements from U_{Q_{1}}(F) ∩ Γ and we have to restrict in all definitions above the groups U_{B}(F) ∩ Γ to its subgroups U_{Q}(F) ∩ Γ for Q ⊇ B. We denote this relation by Opp_{Γ} and define

Opp_{Γ}X_{0} := {Q, Q′) | Q op_{Γ}Q′}.

5. Proof of the theorem (sketch)

In order to define a retraction from the unstable region to its inner boundary, we have to split it up into apartments, thereby constructing a bigger complex (part of an “affine split building”) as follows:

Denote by T , B, Q and P the sets of maximal tori, Borel groups, parabolic and maximal parabolic groups in G, all defined over F (for other notations cf. section 2)

Z := {(x, T) ∈ X′ × T | ∃ B ∈ B : x ∈ D_{B,T} },
by definition D_{B,T} ⊂ X_{T} and T ⊂ B.

Since a maximal torus T is uniquely determined by a pair of opposite Borel groups (B, B′), say T = T_{B,B′}, there exists an equivalent description

Z = {(x, B′) ∈ X′ × B | ∃ B ∈ B : B op B′, x ∈ D_{B,T} for T = T_{B,B′}}

In Z we need an equivalence relation, according to the structure of OppX_{0}, so we define

(x_{1}, T_{1}) ∼ (x_{2}, T_{2}) ⟺ \{ \begin{matrix}x_{1} = x_{2} =: x ∈ D_{B_{1},T_{1}} ∩ D_{B_{2},T_{2}} \\ ∃ Q ∈ Q : Q ⊇ B_{1}, Q ⊇ B_{2}, x ∈ X′′_{Q} \end{matrix}

The group Q is uniquely determined by reduction theory and this fact implies the transitivity of the relation. We can define the equivalence also using the second description of Z:

(x_{1}, B′_{1}) ∼ (x_{2}, B′_{2}) ⟺ \{ \begin{matrix}x_{1} = x_{2} =: x \\ ∃ (Q, Q′) ∈ OppX_{0} : B_{i} ⊆ Q, B′_{i} ⊆ Q′ for i = 1, 2 \\ x ∈ X′′_{Q} \end{matrix}

In this situation the common Levi subgroup L of Q and Q′ is the centralizer of a torus T_{L} (not necessarily maximal), contained in T_{1} ∩ T_{2}. Let us denote by

[x, B′] the class of (x, B′) and by
X̃′ := Z/ ∼= {[x, B′] | (x, B′) ∈ Z} and
X̃′_{Q,Q′} := {[x, B′] ∈ X̃′ | x ∈ X′_{Q}, B′ ⊆ Q′} for (Q, Q′) ∈ Opp X_{0} ,

and finally the analogous definition for X̃′′_{Q,Q′} with x ∈ X′′_{Q}. The topology of X̃′ is given as follows: We choose for X′ the metric topology as a subspace of the affine building X, for T and B the F̂-analytic topology induced from G(F̂), since all maximal tori in T or all Borel groups in B are conjugate under G(F); finally we have the product topology on Z and the quotient topology on X̃′. One should emphasize that every point (x, B′) has an open neighbourhood in Z of the form U × V, where U is the disjoint union of open sets U_{T} in X_{T}, because the complex X is locally finite, so we can avoid ramification inside U_{T}. For a point [x, B′] in X̃′′_{Q,Q′} ⊂ X̃′ there exists a neighbourhood U × V, where U is the union of segments of geodesic lines in X̃′′_{Q,Q′}, defined by the torus T = T_{Δ_{0}} if Q and Q′ are both of cotype Δ_{0}.

We want moreover to define a boundary at infinity for X̃′, generalizing the construction of Landvogt. There the Bruhat–Tits buildings X(Q) := X(Q/R_{u}(Q)), which contribute to the boundary ∂\overline{X} are defined only by quotient groups. For a pair (Q, Q′) of opposite parabolic groups, the common Levi group L = Q ∩ Q′ is isomorphic to Q/R_{u}(Q), so we may consider X(L) instead of X(Q), defined by a subgroup of G. For X̃′ it is more convenient to split up also ∂\overline{X}, using the different buildings X(L) instead of a single X(Q). Therefore we set

∂_{∞}X̃′ := ⋃_{L} X(L) , where L = Q ∩ Q′, (Q, Q′) ∈ Opp X_{0}.
\overline{X}′ := X̃′ ∪ ∂_{∞}X̃′ .

The details are the same as in Landvogt’s construction, but let us remark that for a point of ∂_{∞}X̃′ each neighbourhood meets infinitely many “apartments” X̃′_{T} := {[x, T] ∈ X̃′ | x ∈ X_{T} }. Now we can imitate the proof of proposition 1, in order to determine the homotopy type of X̃′. We have a cover

X̃′ = ⋃_{Opp X_{0}} X̃′_{Q,Q′} = ⋃_{(P,P′)} X̃′_{P,P′} with (P, P′) ∈ Opp X_{0} ∩ (P × P)

with closed sets; their intersections are given by

X̃′_{Q,Q′} = ⋂ {X̃′_{P,P′} | (P, P′) ⊇ (Q, Q′)}

thus this cover has the nerve Opp X_{0}.

The covering sets and their intersections can be surjectively contracted to X(L) ⊂ ∂_{∞}X̃ along geodesic lines defined by the torus T_{L} in the center of L = Q∩Q′ and X(L) is a contractible space, so X̃′_{Q,Q′} is also contractible. Using the result of v. Heydebreck, cited in section 4, we know that Opp X_{0} is (r − 1)-spherical and therefore we have Proposition 2. X̃′ is (r − 1)-spherical.

But in contrast to X′ it is now possible to retract X̃′ to its “inner boundary” (cf. section 2)

Ỹ := ∂_{0}X̃′ := {[x, B′] ∈ X̃′ | x ∈ Y }

along geodesic lines in X̃′′_{Q,Q′}, which do not ramify in X̃′, because we identified different apartments only in these sets X̃′′_{Q,Q′}, and the geodesics coincide in their intersections. Thus we have

Corollary. Ỹ is (r − 1)-spherical.

We need the analogous results for a subcomplex X̃′_{Γ} of X̃′, replacing in the definitions the relation “op” by “op_{Γ}”, consequently we have to admit only pairs of Borel groups (B, B′) with B op_{Γ} B′ and tori T_{B,B′} for (B, B′) ∈ Opp_{Γ}X_{0}. For this purpose we require that also Opp_{Γ}X_{0} is (r − 1)-spherical witch is true for G = SL_{n} by the proof of Charney (see [C]), for the general case see the appendix. Then we obtain

Proposition 3. X̃′_{Γ} and Ỹ_{Γ} := Ỹ ∩ X̃′_{Γ} are (r − 1)-spherical.

The next step is to show that Ỹ_{Γ} is modulo Γ a finite complex — this is the only point where we need X̃′_{Γ} instead of X̃′. For the points of Ỹ_{Γ} the numerical invariants of reduction theory are bounded from above (and below by definition), so part D of the “main theorem” says that Ỹ_{Γ}/Γ is compact. Moreover, by part (E) there exist only finitely many conjugacy classes of Borel groups, therefore in a set of representatives [y, B′] for Ỹ_{Γ}/Γ with y ∈ D_{B,T} only finitely many Borel groups B occur, and since B′ op_{Γ} B, there is only one B′ modulo Γ for each B : Ỹ_{Γ}/Γ is a finite complex. Since all stabilizers in Γ are finite, we can apply the finiteness criterion of K. Brown (see [Br1], 1.1 and 3.1) to get

Proposition 4. Γ is of type F_{r−1}.

Remark to the conjecture “Γ is not of type F_{r}”: Construct an infinite series of (r − 1)-spheres S_{k} in Y = ∂_{0}X′, which are contractible only in growing parts X_{k}, defined by a (rough) filtration of X; then {π_{r−1}(X_{k})} is not “essentially trivial” in the sense of K. Brown (see [Br1], 2).

6. Appendix

For the group G = SL_{n} the complex Opp_{Γ}X_{0} is also (r − 1)-spherical by [C] and so are X̃′_{Γ} and Ỹ_{Γ}. It is not true that Opp_{Γ}X_{0} is a deformation retract of Opp X_{0}, as was shown by Abramenko, who constructed a counter-example. But we have the following

Lemma. X̃′_{Γ} is a deformation retract of X̃′.

Proof: We wish to map a point [x, B′] of X̃′ with x ∈ D_{B,T} ⊂ X_{T}, B op B′, T = T_{B,B′} to [x, B′_{0}] with the same x ∈ X′ and B′_{0} op_{Γ} B, obtaining a new torus T_{0} := T_{B,B′_{0}}. Identifying the Borel groups opposite to B with elements of U(F) (the unipotent radical of B), for [x, B′] the group B′ corresponds to an element of U(F) ∩ K_{x} with K_{x} = Stab_{G(F)}(x) since x ∈ X_{T_{B,B′}}. This compact group contains finitely many elements of the discrete group U(F) ∩ Γ; we have to make a choice: There is one element, defining B′_{0} and T_{0}, such that X_{T_{0}} ∩X_{T} is maximal because the intersection is given as the intersection of half-apartments, defined by root groups, and for a Chevalley group, U is the semi-direct product of its root-groups. This definition is compatible with the equivalence relation in Z and the map induces the identity on X̃′_{Γ}. This map is also continuous: The topology in the second component is induced by the analytic topology of the group G(F̂); an element of U(F)∩K_{x} has a neighbourhood which contains only one element of U(F) ∩ Γ, due to its discreteness.

Remark. Since Opp_{Γ}X_{0} is the nerve of a cover of X̃′_{Γ}, we proved indirectly that it is (r − 1)-spherical. A direct proof for groups over Dedekind rings would be of interest.

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Frankfurt a.M., February 7, 2001