COHOMOLOGY OF BUILDINGS AND FINITENESS PROPERTIES OF Ã_{n}-GROUPS

JACQUI RAMAGGE AND WAYNE W. WHEELER

Abstract. Borel and Serre calculated the cohomology of the building associated to a reductive group and used the result to deduce that torsion-free S-arithmetic groups are duality groups. By replacing their group-theoretic arguments with proofs relying only upon the geometry of buildings, we show that Borel and Serre’s approach can be modified to calculate the cohomology of any locally finite affine building. As an application we show that any finitely presented Ã_{n}- group is a virtual duality group. A number of other finiteness conditions for Ã_{n}-groups are also established.

1. Introduction

An important result of Borel and Serre shows that a torsion-free S-arithmetic group Γ is a duality group [BS]. In other words, there exist a natural number d and a ℤΓ-module D such that for every integer i there is a natural isomorphism

H^{i}(Γ, −) ≅ H_{d−i}(Γ, D ⊗ −).

Borel and Serre obtain this result by first studying the building associated to a reductive group G. In particular, they calculate the Alexander–Spanier cohomology of the building and deduce that a torsion-free S-arithmetic subgroup of G is a duality group.

The purpose of this paper is to show that the ideas of Borel and Serre, which seem to depend upon properties of the group G, can be adapted so that they apply to other situations as well. In particular, we calculate the cohomology of an arbitrary locally finite affine building Δ. Since Δ need not arise from an associated reductive group, the arguments given here are necessarily more geometric than those used by Borel and Serre. Our hope is that an approach relying only upon the geometry of buildings will be accessible to a wider audience than one relying upon the properties of reductive groups. At the same time, the ideas presented here should help to make it clearer which arguments in [BS] depend fundamentally upon the group G and which arguments depend only upon properties of the associated building.

As an application of the result describing the cohomology of Δ, we study the homological finiteness properties of certain groups acting on buildings. Groups that act simply transitively on the vertices of a thick Ã_{2}-building in a type-rotating manner, known as Ã_{2}-groups, were introduced in [CMSZ1] and [CMSZ2]. The more general notion of an Ã_{n}-group was studied in [C]. Any such group has a relatively simple presentation with respect to which the Cayley

2 JACQUI RAMAGGE AND WAYNE W. WHEELER

graph is the 1-skeleton of an Ã_{n}-building. One of the reasons why such groups are of interest is that for n = 2 they give rise to buildings that cannot be obtained from classical groups. We show that any finitely generated Ã_{n}-group Γ is a virtual duality group; in other words, Γ has a subgroup K of finite index that is a duality group. The argument proving this result applies uniformly to all Ã_{n}-groups, but a shorter proof is possible in some cases. For n > 2 we show that the result can be obtained as a consequence of the work in [BS]; for n = 2 we include an argument of G. Robertson using Kazhdan’s Property (T) to prove that certain Ã_{2}-groups are virtual duality groups.

The broad outline of the paper generally follows the proof of Borel and Serre that torsionfree S-arithmetic groups are duality groups. In particular, Section 3 constructs a compact space X̄ in which the locally finite affine building Δ embeds as an open subspace. Section 4 develops geometric arguments to calculate the Alexander–Spanier cohomology with compact supports H_{c}^{∗}(Δ; ℤ). In Section 5 the cohomology H_{c}^{∗}(Δ; ℤ) is used to show that finitely presented Ã_{n}-groups are virtual duality groups, and a number of other finiteness conditions are also given.

The authors are indebted to John Crisp for helpful discussions on the material in Section 3. The work presented here was partially supported by the University of Newcastle and the University of Leicester, and the second author was partially supported by the National Science Foundation. The authors are grateful to all of these organizations for their assistance.

2. Background material

The purpose of this section is to summarize some definitions, properties, and notational conventions for buildings and for Alexander–Spanier cohomology that are used in the remainder of the paper. The primary references for buildings are [B2] and [R]; a good reference for Alexander–Spanier cohomology is [S].

Recall that a building (Δ, A) is a simplicial complex Δ and a collection A of subcomplexes called apartments that satisfy the following axioms:

(1) Δ = ⋃_{A∈A} A;

(2) every apartment A ∈ A is a Coxeter complex;

(3) for any two simplices σ_{1}, σ_{2} ∈ Δ there is an apartment A containing both σ_{1} and σ_{2};

(4) if A and A′ are two apartments containing simplices σ_{1} and σ_{2}, then there is an isomorphism A → A′ fixing σ_{1} and σ_{2} pointwise.

The conventions used throughout this paper are generally the same as those used in [B2]. In particular, we regard chambers as open subsets of the geometric realization of Δ. Because we

COHOMOLOGY OF BUILDINGS AND FINITENESS PROPERTIES OF Ã_{n}-GROUPS 3

are usually more concerned with the geometric realization of Δ than with Δ itself, we have chosen also to write Δ for the geometric realization of the building when no confusion is likely to arise.

If Δ is a building, then the apartments of Δ are all isomorphic. For any apartment A there is a Coxeter group (W, S) such that A is isomorphic to the Coxeter complex Σ(W, S), and this Coxeter group gives a corresponding W -valued distance function on the set of chambers of A. If C is the set of chambers of Δ, then this function extends to a W -valued distance function δ : C × C → W. When W is finite, the building Δ is said to be spherical; otherwise, Δ is said to be affine. This terminology reflects the structure of the geometric realizations of apartments, which are homeomorphic to either n-spheres or affine n-space for some fixed n.

An affine building Δ is a path-metric space with unique geodesics. Moreover, Δ satisfies the CAT(0)-property. In other words, if x, y ∈ Δ and 0 ≤ t ≤ 1, then the point z = (1 − t)x + ty on the geodesic [x, y] satisfies the inequality

d(w, z)^{2} ≤ (1 − t)d(w, x)^{2} + td(w, y)^{2} − t(1 − t)d(x, y)^{2}

for all w ∈ Δ (see Lecture 4 of [B3]).

If Δ is an affine building, then a standard construction produces another building Δ_{∞}, called the spherical building at infinity. A ray in Δ is a subset 𝔯 that is isometric to the half-line [0, ∞); the points of Δ_{∞} are the equivalence classes of rays under translation. Let x_{0} be a special vertex in Δ. Then the points in Δ_{∞} are in one-to-one correspondence with the rays based at x_{0}, and the chambers in Δ_{∞} correspond to sectors based at x_{0}.

Throughout this work we write simply H^{i}(X; ℤ) for the ith Alexander–Spanier cohomology group of a space X because that is the only cohomology theory used; similarly, we write H_{c}^{i}(X; ℤ) for the ith Alexander–Spanier cohomology group of X with compact supports, and H̃^{i}(X; ℤ) denotes the reduced ith Alexander–Spanier cohomology group of X. In particular, H^{0}(X; ℤ) is isomorphic to the group of locally constant functions from X to ℤ, and H_{c}^{0}(X; ℤ) is isomorphic to the subgroup of locally constant functions with compact supports. The reduced Alexander–Spanier cohomology agrees with the ordinary Alexander–Spanier cohomology in positive degrees and satisfies H̃^{0}(X; ℤ) = 0 if X is connnected. We use a number of properties of Alexander–Spanier cohomology in the following sections, and the reader is encouraged to consult [S] or [Ma] for further details.

3. An analogue of the Borel–Serre space

In [BS] Borel and Serre studied the building Δ of a reductive group by constructing a compact space X̄ and a homeomorphism ϕ from Δ onto an open subspace of X̄ such that the points

4 JACQUI RAMAGGE AND WAYNE W. WHEELER

of X_{∞} = X̄ − ϕ(Δ) are in one-to-one correspondence with those of the building at infinity Δ_{∞}. The purpose of this section is to construct and study the space X̄ when Δ is an arbitrary locally finite affine building. It is necessary, therefore, to replace the group-theoretic arguments of Borel and Serre by geometric proofs. These proofs rely heavily upon the fact that Δ is a complete metric space. But the constructions of this section work somewhat more generally: it is only necessary to assume that Δ is a complete, locally compact CAT(0)-space with a metric d, and we will work in this more general setting throughout this section.

Fix a point x_{0} ∈ Δ. Let X denote the collection of all geodesics [x_{0}, x] for x ∈ Δ, and let X_{∞} denote the collection of all geodesic rays [x_{0}, e) based at x_{0}. A geodesic [x_{0}, x] of length s can be regarded as a function γ_{x} : [0, ∞) → Δ that is an isometry on [0, s] and is constant on [s, ∞); similarly, a geodesic ray [x_{0}, e) can be regarded as an isometry on its entire domain. Thus X and X_{∞} are disjoint subsets of the collection C([0, ∞), Δ) of continuous functions from [0, ∞) to Δ. The topology of compact convergence on C([0, ∞), Δ) then gives the subspace X̄ = X ∪ X_{∞} a topological structure. If K ⊆ [0, ∞) is compact, γ : [0, ∞) → Δ is in X̄, and ε > 0, then the set

B_{K}(γ, ε) = {γ′ ∈ X̄ | d(γ(t), γ′(t)) < ε for all t ∈ K}

is open in X̄. In fact, the sets of this form are a basis for the topology on X̄. By using the properties of CAT(0)-spaces, however, one can simplify this description with the following lemma.

Lemma 3.1. Let Δ be a complete, locally compact CAT(0)-space. Suppose that y, z ∈ Δ are distinct points such that d(x_{0}, y) = d(x_{0}, z) > 0, and assume that y′ ∈ [x_{0}, y] and z′ ∈ [x_{0}, z] satisfy d(x_{0}, y′) = d(x_{0}, z′). Then d(y′, z′) ≤ d(y, z); equality holds if and only if y′ = y and z′ = z.

Proof. Let s = d(x_{0}, y) = d(x_{0}, z) and s′ = d(x_{0}, y′) = d(x_{0}, z′). Then t = s′/s satisfies 0 ≤ t ≤ 1; moreover, y′ = (1 − t)x_{0} + ty and z′ = (1 − t)x_{0} + tz. The CAT(0)-inequality implies that

d(y′, z′)^{2} ≤ (1 − t)d(y′, x_{0})^{2} + td(y′, z)^{2} − t(1 − t)d(x_{0}, z)^{2}

and that

d(z, y′)^{2} ≤ (1 − t)d(z, x_{0})^{2} + td(z, y)^{2} − t(1 − t)d(x_{0}, y)^{2}.

Hence

d(y′, z′)^{2} ≤ (1 − t)(s′)^{2} + t^{2}d(y, z)^{2} − t^{2}(1 − t)s^{2} = t^{2}d(y, z)^{2}.

COHOMOLOGY OF BUILDINGS AND FINITENESS PROPERTIES OF Ã_{n}-GROUPS 5

Thus d(y′, z′) ≤ td(y, z) ≤ d(y, z), and d(y′, z′) = d(y, z) if and only if t = 1. The desired result now follows immediately.

To apply Lemma 3.1, consider a basic open set B_{K}(γ, ε) in X̄. Because K is a compact subset of [0, ∞), it contains a largest element M. Then K ⊆ [0, M], and Lemma 3.1 implies that B_{K}(γ, ε) = B_{[0,M]}(γ, ε). We can therefore simplify the notation by writing B_{M}(γ, ε) for B_{[0,M]}(γ, ε), and the sets B_{M}(γ, ε) with γ ∈ X̄ and M and ε positive form a basis for the topology on X̄.

Lemma 3.2. Let Δ be a complete, locally compact CAT(0)-space, and let S be a subset of Δ. Then S is compact if and only if it is closed and bounded.

Proof. Fix x ∈ S, and set B_{r}(x) = {y ∈ Δ | d(x, y) < r}. Then {B_{r}(x) ∩ S | r > 0} is an open cover of S. If S is compact, then it follows that S ⊆ B_{r}(x) for some r > 0, so S is bounded. Because Δ is Hausdorff, S is also closed.

Conversely, suppose that S is closed and bounded in Δ. Choose r > 0 such that S ⊆ B_{r}(x). Then the closure B̄_{r}(x) is compact by Proposition 2.3 of [P], so S is also compact, as desired.

For r > 0 set B_{r} = {x ∈ Δ | d(x_{0}, x) < r}. Let B̄_{r} be the closure of B_{r} in Δ, and set ∂B̄_{r} = B̄_{r} −B_{r}. Let ϕ : Δ → X̄ be the function given by ϕ(x) = [x_{0}, x]. It is easy to check that ϕ is a homeomorphism of Δ onto X, so we can regard Δ as an open subspace of X̄. Moreover, the subspace X_{∞} ⊆ X̄ is compact by Ascoli’s Theorem (Theorem 7.6.1 of [Mu]).

Proposition 3.3. Let Δ be a complete, locally compact CAT(0)-space. Then the space X̄ is compact.

Proof. It suffices to show that if U is a covering of X̄ by basic open sets, then U has a finite subcover. Because X_{∞} is compact, there are open sets U_{1}, …, U_{k} ∈ U such that {U_{1} ∩ X_{∞}, …, U_{k} ∩ X_{∞}} is an open cover of X_{∞}. For 1 ≤ i ≤ k write U_{i} = B_{r_{i}}(γ_{i}, ε_{i}) for some γ_{i} ∈ X̄ and some positive numbers r_{i} and ε_{i}. Set r = max{r_{1}, …, r_{k}}, and suppose that x ∈ Δ satisfies d(x_{0}, x) = s > r. The geodesic [x_{0}, x] in Δ is the initial segment of some ray [x_{0}, e) in X_{∞}, and the choice of U_{1}, …, U_{k} implies that there is an i with [x_{0}, e) ∈ U_{i} ∩X_{∞}. Then ϕ(x) = [x_{0}, x] ∈ U_{i}, and we conclude that

X̄ − (U_{1} ∪ ⋯ ∪ U_{k}) ⊆ ϕ(B̄_{r}).

But ϕ(B̄_{r}) is compact by Lemma 3.2, so there are open sets U_{k+1}, …, U_{m} ∈ U such that ϕ(B̄_{r}) ⊆ U_{k+1} ∪ ⋯ ∪ U_{m}. Then {U_{1}, …, U_{m}} is a finite subcover of X̄, and X̄ is compact. This completes the proof.

6 JACQUI RAMAGGE AND WAYNE W. WHEELER

Theorem 3.4. Let Δ be a complete, locally compact CAT(0)-space. Then the topology T on X̄ has the following properties:

(1) the space X̄ is compact and contractible;

(2) X is open and X_{∞} is closed in X̄;

(3) Δ is homeomorphic to X.

Proof. Proposition 3.3 shows that X̄ is compact. To see that it is contractible, suppose that γ ∈ X̄ and t ∈ [0, 1]. Let γ_{t} ∈ X̄ be the geodesic satisfying the following conditions:

(a) if γ = ϕ(x) for some x ∈ Δ and x_{t} ∈ Δ is the unique point on the geodesic γ with

d(x_{0}, x_{t}) = \frac{td(x_{0}, x)}{1 + (1 − t)d(x_{0}, x)}, then γ_{t} = [x_{0}, x_{t}];

(b) if γ ∈ X_{∞} and t < 1, then γ_{t} = [x_{0}, x_{t}] for the unique point x_{t} on the geodesic γ satisfying d(x_{0}, x_{t}) = \frac{t}{1 − t};

(c) if γ ∈ X_{∞} and t = 1, then γ_{t} = γ. Define F : X̄ × [0, 1] → X̄ by setting F (γ, t) = γ_{1−t}. Then F is continuous and is a contraction of X̄ onto γ_{0} = [x_{0}, x_{0}], so (1) holds. The remaining statements follow easily from the definition of T , and this completes the proof.

4. The cohomology of an affine building

If Δ is a locally finite affine building, then the previous section described a compact space X̄ such that the points of X_{∞} are canonically identified with those of the building Δ_{∞}. In this section we calculate H^{∗}(X_{∞}; ℤ) and use the result to determine the cohomology H_{c}^{∗}(Δ; ℤ). The arguments are analogous to those used by Borel and Serre [BS], who studied the building associated to a reductive group. By avoiding their group-theoretic arguments, however, we obtain a more general result that verifies an assertion of Brown in Lecture 5 of [B3].

Throughout this section let Δ be a locally finite affine building. It will be convenient to assume that the fixed base point x_{0} ∈ Δ is a special vertex. Then the collection ℭ of all chambers in Δ_{∞} can be identified with the set S of open sectors in Δ based at x_{0}. If 𝔠_{1}, 𝔠_{2} ∈ ℭ are chambers represented by sectors σ_{1}, σ_{2} ∈ S, then there are unique chambers C_{1} ⊆ σ_{1} and C_{2} ⊆ σ_{2} of Δ with x_{0} ∈ C̄_{1} ∩ C̄_{2}; moreover, 𝔠_{1} and 𝔠_{2} are distinct and adjacent if and only if C_{1} and C_{2} are distinct and adjacent, and in this case the adjacency type of C_{1} and C_{2} determines that of 𝔠_{1} and 𝔠_{2}.

Although X_{∞} and Δ_{∞} are equal as sets, the topology on X_{∞} is not the usual simplicial topology on Δ_{∞} in which a set is open if and only if its intersection with every closed simplex is relatively open. For example, if Δ is a tree, then Δ_{∞} is a discrete space, whereas X_{∞} is compact by Theorem 3.4. Thus X_{∞} and Δ_{∞} must be different topological spaces if Δ has

COHOMOLOGY OF BUILDINGS AND FINITENESS PROPERTIES OF Ã_{n}-GROUPS 7

infinitely many ends. The fact that the topologies on X_{∞} and on Δ_{∞} can differ presents a minor technical problem: if 𝔠 is a chamber of Δ_{∞} and 𝔠̄ is its closure in Δ_{∞}, then it is perhaps not obvious that 𝔠̄ is also closed in X_{∞}. We begin by proving this result.

Proposition 4.1. Let 𝔠 be a chamber in Δ_{∞}, and let 𝔠̄ be its closure in Δ_{∞}. Then 𝔠̄ is also closed in X_{∞}.

Proof. Let σ ∈ S be the sector representing 𝔠, and let σ̄ be the closure of σ in Δ. Then 𝔠̄ is just the collection of all geodesic rays based at x_{0} that are contained in σ̄. Thus if [x_{0}, e) ∈ X_{∞} − 𝔠̄, then there is a point y ∈ [x_{0}, e)−σ̄. Choose ε > 0 such that the open ball in Δ with radius ε and center y does not intersect σ̄, and set r = d(x_{0}, y). Then B_{r}([x_{0}, e), ε) is an open neighborhood of [x_{0}, e) in X̄ that does not intersect 𝔠̄. Hence 𝔠̄ is closed in X_{∞}, as desired.

If 𝔠 is a chamber in Δ_{∞}, then Proposition 4.1 implies that its closure in Δ_{∞} is the same as its closure in X_{∞}. Thus we can refer unambiguously to its closure 𝔠̄.

Now fix an apartment A_{0} in Δ containing x_{0}, and let 𝔞_{0} be the corresponding apartment in Δ_{∞}. Then there is a finite Coxeter group (W, S) such that 𝔞_{0} is isomorphic to the Coxeter complex of (W, S). If w ∈ W, let ℓ(w) be the length of w. Write W = {w_{1}, …, w_{N}}, where the indices are chosen so that ℓ(w_{i}) ≤ ℓ(w_{j}) whenever i ≤ j. In particular, w_{1} = 1. For 1 ≤ m ≤ N let

I_{m} = {s ∈ S | ℓ(w_{m}s) > ℓ(w_{m})}.

Then I_{1} = S, I_{N} = ∅, and I_{m} is a nonempty proper subset of S for 1 < m < N. Let δ be the W -valued distance function defined on the set of chambers of Δ_{∞}, and let 𝔠_{0} be a fixed chamber in 𝔞_{0}. For each element w_{i} ∈ W let ℭ_{i} be the set of all chambers 𝔠 in Δ_{∞} such that δ(𝔠_{0}, 𝔠) = w_{i}, and let X_{m} be the union of closed chambers defined by

X_{m} = ⋃_{i=1}^{m} ⋃_{𝔠∈ℭ_{i}} 𝔠̄.

Then 𝔠̄_{0} = X_{1} ⊆ X_{2} ⊆ ⋯ ⊆ X_{N} = X_{∞}. Set

L_{m} = ⋃_{I⊆I_{m}}(𝔠_{0})_{I},

the union of the I-residues of 𝔠_{0}. The set L_{m} is open in 𝔠̄_{0}. Indeed, the complement L′_{m} = 𝔠̄_{0} − L_{m} is the union of the closures of the faces (𝔠_{0})_{{s}} of codimension one for s ∈ S − I_{m}.

Example 4.2. We illustrate the above definitions when X_{∞} is the building of PGL_{3}(𝔽_{2}), pictured below with a distinguished chamber 𝔠_{0}. In this case X_{∞} is the building at infinity for a suitable affine building Δ that is neither thick nor homogeneous.

• ∘ • ∘ • ∘ • ∘  ∘ • ∘ • ∘ •  𝔠_{0}  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The Coxeter group W is a symmetric group on three letters; it has a generating set S = {s_{1}, s_{2}}, where s_{i} denotes the transposition (i, i + 1). The solid nodes in the diagram represent s_{1}- adjacency, and the circles represent s_{2}-adjacency. We order the elements of W by setting

w_{1} = 1, w_{2} = s_{1}, w_{3} = s_{2}, w_{4} = s_{1}s_{2}, w_{5} = s_{2}s_{1}, w_{6} = s_{1}s_{2}s_{1} = s_{2}s_{1}s_{2}.

Then

I_{1} = {s_{1}, s_{2}}, I_{2} = I_{5} = {s_{2}}, I_{3} = I_{4} = {s_{1}}, I_{6} = ∅,

L_{1} = •  ∘_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} , L_{2} = L_{5} =  ∘_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} , L_{3} = L_{4} = •_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} , and L_{6} = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

COHOMOLOGY OF BUILDINGS AND FINITENESS PROPERTIES OF Ã_{n}-GROUPS 9

Moreover, ℭ_{1} = {𝔠_{0}}, and in the following diagrams the solid lines represent the chambers that are elements of the set ℭ_{i} for 2 ≤ i ≤ 6.

• ∘ • ∘ • ∘ • ∘  ∘ • ∘ • ∘ •  ℭ_{2}  _{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}  • ∘ • ∘ • ∘ • ∘  ∘ • ∘ • ∘ •  ℭ_{3}  _{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

• ∘ • ∘ • ∘ • ∘  ∘ • ∘ • ∘ •  ℭ_{4}  _{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  • ∘ • ∘ • ∘ • ∘  ∘ • ∘ • ∘ •  ℭ_{5}  _{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

• ∘ • ∘ • ∘ • ∘  ∘ • ∘ • ∘ •  ℭ_{6}  _{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}

Then X_{1} = 𝔠̄_{0} and X_{6} = X_{∞}, and the sets X_{i} for 2 ≤ i ≤ 5 are given in the following diagrams.

• ∘ • ∘ • ∘ • ∘  ∘ • ∘ • ∘ •  X_{2}  _{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  • ∘ • ∘ • ∘ • ∘  ∘ • ∘ • ∘ •  X_{3}  _{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.} _{.}  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Observe that for 2 ≤ m ≤ 6 the set X_{m} − X_{m−1} is a disjoint union of subspaces homeomorphic to L_{m}, with one copy of L_{m} for each chamber 𝔠 ∈ ℭ_{m}. This is a general phenomenon, as Proposition 4.5 below shows.

In order to obtain a general description of the spaces X_{m} − X_{m−1}, we begin by defining a topology on the set S of all sectors in Δ based at x_{0}. The topology has a basis given by sets of the form

N(σ, r) = {σ′ ∈ S | σ′ ∩ B̄_{r} = σ ∩ B̄_{r}}

for σ ∈ S and r > 0. Because S can be identified with the collection ℭ of all chambers in Δ_{∞}, this definition also provides a topology on ℭ and on the subspaces ℭ_{i} ⊆ ℭ for 1 ≤ i ≤ N. If 𝔠 ∈ ℭ is a chamber represented by the sector σ ∈ S, then we write N(𝔠, r) for the basic open neighborhood of 𝔠 consisting of all chambers represented by sectors in N(σ, r). The topological space S has previously been used by analysts, who usually denote it by Ω and call it the boundary of the building Δ (for example, see [MZ] or [RR]).

Proposition 4.3. The space S is compact and totally disconnected. Moreover, ℭ_{i} is compact and totally disconnected for 1 ≤ i ≤ N.

Proof. The proof of the second statement is virtually identical to that of the first, so we prove the result only for S. Let σ_{1} and σ_{2} be distinct sectors in S. Then there is an r > 0 such that σ_{1} ∩ B̄_{r} ≠ σ_{2} ∩ B̄_{r}. Since Δ is locally finite, {σ_{1}, σ_{2}} can be extended to a finite collection of sectors {σ_{1}, …, σ_{t}} such that if σ ∈ S, then there is a unique i with 1 ≤ i ≤ t satisfying σ ∩ B̄_{r} = σ_{i} ∩ B̄_{r}. Set U_{1} = N(σ_{1}, r) and U_{2} = N(σ_{2}, r) ∪ ⋯ ∪ N(σ_{t}, r). Then U_{1}

12 JACQUI RAMAGGE AND WAYNE W. WHEELER

and U_{2} are disjoint open subsets of S such that U_{1} ∪ U_{2} = S. Thus σ_{1} and σ_{2} lie in different components of S, and S is totally disconnected.

For each i ∈ ℕ let F_{i} be the collection of subsets of ∂B̄_{i} of the form σ ∩ ∂B̄_{i} for some σ ∈ S. Then F_{i} is a finite set, so it is a compact topological space with the discrete topology. The space ∏_{i=1}^{∞} F_{i} is again compact, and there is a map ψ : S → ∏_{i=1}^{∞} F_{i} given by ψ(σ) = (σ ∩ ∂B̄_{i})_{i∈ℕ}. It is straightforward to check that ψ maps S homeomorphically onto a closed subset. Thus S is compact, as desired.

Proposition 4.4. The sets X_{m} are closed in X_{∞} for 1 ≤ m ≤ N.

Proof. Let [x_{0}, e) be any point in X_{∞} − X_{m}, and let R ⊆ S be the collection of all sectors that represent a chamber 𝔠 in Δ_{∞} with [x_{0}, e) ∈ 𝔠̄. Such a chamber must satisfy δ(𝔠_{0}, 𝔠) = w_{i} for some i > m. Let y ≠ x_{0} be a point on the geodesic [x_{0}, e), and set s = d(x_{0}, y). Choose ε > 0 so that the open ball B_{ε}(y) satisfies

B_{ε}(y) ⊆ ⋃_{σ∈R} σ̄.

Then U = B_{s}([x_{0}, e), ε) ∩ X_{∞} is an open neighborhood of [x_{0}, e) in X_{∞} that is contained in the union of all closed chambers 𝔠̄ with [x_{0}, e) ∈ 𝔠̄.

Suppose that U ∩ X_{m} ≠ ∅. Then there exist an i ≤ m and a chamber 𝔠_{i} ∈ ℭ_{i} such that U ∩ 𝔠̄_{i} ≠ ∅. But U is open, so U ∩ 𝔠_{i} ≠ ∅. Since U is contained in the union of all closed chambers 𝔠̄ with [x_{0}, e) ∈ 𝔠̄, we obtain a contradiction. Hence U ∩ X_{m} = ∅. It follows that X_{∞} − X_{m} is open so that X_{m} is closed, as desired.

Proposition 4.5. If 1 < m ≤ N, then X_{m} − X_{m−1} is homeomorphic to ℭ_{m} × L_{m}.

Proof. For any chamber 𝔠 ∈ ℭ let ϕ_{𝔠_{0}𝔠} : 𝔠̄_{0} → 𝔠̄ be the geometric realization of the unique type-preserving simplicial map from 𝔠_{0} to 𝔠. If 𝔠 ∈ ℭ_{m}, then ϕ_{𝔠_{0}𝔠} restricts to a homeomorphism between L_{m} and (X_{m} − X_{m−1}) ∩ 𝔠̄. Define Φ : ℭ_{m} × L_{m} → X_{m} − X_{m−1} by

Φ(𝔠, γ) = ϕ_{𝔠_{0}𝔠}(γ).

We claim that Φ is a homeomorphism.

If γ ∈ X_{m} − X_{m−1}, then γ ∈ 𝔠̄ for some chamber 𝔠 ∈ ℭ_{m}, and γ ∉ 𝔡̄ for any chamber 𝔡 ∈ ℭ_{i} with i < m. Thus there is a γ_{0} ∈ L_{m} with γ = ϕ_{𝔠_{0}𝔠}(γ_{0}) = Φ(𝔠, γ_{0}), and Φ is onto. Moreover, if γ = Φ(𝔠, γ_{0}) = Φ(𝔠′, γ′_{0}), then γ ∈ 𝔠̄ ∩ 𝔠̄′. Let 𝔟 be a common face of 𝔠 and 𝔠′, and let 𝔞_{0} be an apartment in Δ_{∞} containing 𝔠_{0} and 𝔟. Because γ ∉ X_{m−1}, any minimal gallery from 𝔠_{0} to 𝔠 is also a minimal gallery stretched from 𝔠_{0} to 𝔟, so it is contained in 𝔞_{0}. In particular, 𝔠 is a chamber in 𝔞_{0}. The same argument shows that 𝔠′ is also in 𝔞_{0}, and δ(𝔠_{0}, 𝔠) = δ(𝔠_{0}, 𝔠′) = w_{m}. Then 𝔠 = 𝔠′ and γ_{0} = γ′_{0}, so Φ is a one-to-one correspondence.

COHOMOLOGY OF BUILDINGS AND FINITENESS PROPERTIES OF Ã_{n}-GROUPS 13

Before proving that Φ and Φ^{−1} are continuous, it is useful to make a preliminary observation: if γ ∈ 𝔠̄_{0}, 𝔠 and 𝔠′ are chambers of Δ_{∞} with 𝔠′ ∈ N(𝔠, r), and y is the point on the geodesic ϕ_{𝔠_{0}𝔠}(γ) ∈ X_{∞} with d(x_{0}, y) = r, then y also lies on the geodesic ϕ_{𝔠_{0}𝔠′}(γ). Indeed, let σ ∈ S be the sector corresponding to 𝔠, and let σ′ ∈ S be the sector corresponding to 𝔠′. Choose apartments A and A′ in Δ with σ̄ ⊆ A and σ̄′ ⊆ A′. There is a unique type-preserving isometry ψ : A → A′ fixing A ∩ A′ pointwise, and ψ induces a map ψ_{∞} : A_{∞} → A′_{∞} on the corresponding apartments in Δ_{∞}. This map satisfies ψ_{∞}(𝔠) = 𝔠′ because σ′ ∩ B̄_{r} = σ ∩ B̄_{r}. Thus ψ_{∞}|𝔠̄ = ϕ_{𝔠_{0}𝔠′} ∘ϕ_{𝔠_{0}𝔠}^{−1}, and ϕ_{𝔠_{0}𝔠′}(γ) = ψ_{∞}ϕ_{𝔠_{0}𝔠}(γ). Moreover, [x_{0}, y] ⊆ A∩A′, so ψ fixes [x_{0}, y]. It follows that [x_{0}, y] is the initial segment of the geodesics ϕ_{𝔠_{0}𝔠}(γ) and ϕ_{𝔠_{0}𝔠′}(γ), as claimed.

To prove that Φ is continuous, let B_{r}(γ_{0}, ε) be a basic open set in X̄, and set

B̂_{r}(γ_{0}, ε) = B_{r}(γ_{0}, ε) ∩ (X_{m} − X_{m−1}).

It suffices to show that Φ^{−1}(B̂_{r}(γ_{0}, ε)) is open in ℭ_{m}×L_{m}. Suppose that (𝔠, γ) ∈ Φ^{−1}(B̂_{r}(γ_{0}, ε)). Define an open subset V ⊆ L_{m} by setting V = ϕ_{𝔠_{0}𝔠}^{−1}(B̂_{r}(γ_{0}, ε)). Then (N(𝔠, r) ∩ ℭ_{m}) × V is an open neighborhood of (𝔠, γ) in ℭ_{m} × L_{m}, and we claim that it is contained in Φ^{−1}(B̂_{r}(γ_{0}, ε)). Indeed, suppose that (𝔠′, γ′) ∈ (N(𝔠, r) ∩ ℭ_{m}) × V, and let y ∈ Δ be the point on the geodesic ϕ_{𝔠_{0}𝔠}(γ′) ∈ X_{∞} with d(x_{0}, y) = r. Then [x_{0}, y] is also the initial segment of the geodesic ϕ_{𝔠_{0}𝔠′}(γ′). Because γ′ ∈ V, we have d(γ_{0}(r), y) < ε. Then Φ(𝔠′, γ′) = ϕ_{𝔠_{0}𝔠′}(γ′) ∈ B̂_{r}(γ_{0}, ε), as desired.

Finally, we must show that Φ maps open subsets of ℭ_{m} ×L_{m} to open subsets of X_{m} −X_{m−1}. It suffices to prove this result for an open set of the form N(𝔠, r)×(B_{r′}(γ, ε)∩L_{m}) with 𝔠 ∈ ℭ_{m} and γ ∈ L_{m}. Note that the image of this set can be written as a disjoint union:

Φ(N(𝔠, r) × (B_{r′}(γ, ε) ∩ L_{m})) = ⋃_{𝔠′∈N(𝔠,r)}ϕ_{𝔠_{0}𝔠′}(B_{r′}(γ, ε) ∩ L_{m})

= ⋃_{𝔠′∈N(𝔠,r)}B_{r′}(ϕ_{𝔠_{0}𝔠′}(γ), ε) ∩ ϕ_{𝔠_{0}𝔠′}(L_{m})

= ⋃_{𝔠′∈N(𝔠,r)}B_{r′}(ϕ_{𝔠_{0}𝔠′}(γ), ε) ∩ (X_{m} − X_{m−1}) ∩ 𝔠̄′.

Assume that r′ ≤ r. Then B_{r′}(ϕ_{𝔠_{0}𝔠′}(γ), ε) = B_{r′}(ϕ_{𝔠_{0}𝔠}(γ), ε) for all 𝔠′ ∈ N(𝔠, r) ⊆ N(𝔠, r′), so

Φ(N(𝔠, r) × (B_{r′}(γ, ε) ∩ L_{m})) = B_{r′}(ϕ_{𝔠_{0}𝔠}(γ), ε) ∩ ⋃_{𝔠′∈N(𝔠,r)}(X_{m} − X_{m−1}) ∩ 𝔠̄′
= B_{r′}(ϕ_{𝔠_{0}𝔠}(γ), ε) ∩ Φ(N(𝔠, r) × L_{m}),

and it suffices to prove that Φ(N(𝔠, r) × L_{m}) is open. Suppose that γ′ ∈ (X_{m} − X_{m−1}) ∩ 𝔠̄′ for some 𝔠′ ∈ N(𝔠, r). If there is a chamber 𝔠′′ ≠ 𝔠′ such that γ′ ∈ 𝔠̄′′, then δ(𝔠_{0}, 𝔠′′) = w_{k} for some k ≥ m because γ′ ∈ X_{m} − X_{m−1}; in fact, k > m because Φ is one-to-one. If there is no such chamber 𝔠′′, then γ′ lies in the interior 𝔠′ of 𝔠̄′. In either case there is a number ε′ > 0 such that

B_{r}(γ′, ε′) ∩ X_{∞} ⊆ ( ⋃_{𝔠′∈N(𝔠,r)} 𝔠̄′) ∪ ( ⋃_{k=m+1}^{N} ⋃_{𝔠′′∈ℭ_{k}} 𝔠̄′′).

Then γ′ ∈ B_{r}(γ′, ε′) ∩ (X_{m} − X_{m−1}) ⊆ Φ(N(𝔠, r) × L_{m}), so Φ(N(𝔠, r) × L_{m}) is open in X_{m} − X_{m−1}, as desired.

Now suppose that r < r′, and let σ ∈ S be the sector corresponding to the chamber 𝔠. Because Δ is locally finite, there exists a finite set of sectors {σ_{1}, …, σ_{t}} ⊆ N(σ, r) such that if σ′ ∈ N(σ, r), then σ′ ∩ B̄_{r′} = σ_{i} ∩ B̄_{r′} for a unique i with 1 ≤ i ≤ t. Let 𝔠_{i} ∈ N(𝔠, r) be the chamber corresponding to σ_{i}. Then

Φ(N(𝔠, r) × (B_{r′}(γ, ε) ∩ L_{m})) = ⋃_{i=1}^{t} ⋃_{𝔠′∈N(𝔠_{i},r′)}B_{r′}(ϕ_{𝔠_{0}𝔠′}(γ), ε) ∩ (X_{m} − X_{m−1}) ∩ 𝔠̄′

= ⋃_{i=1}^{t} Φ(N(𝔠_{i}, r′) × (B_{r′}(γ, ε) ∩ L_{m})),

and this set is open in X_{m} − X_{m−1} by the previous paragraph. Thus Φ is a homeomorphism, and this completes the proof.

We are now able to calculate the Alexander–Spanier cohomology H^{∗}(X_{∞}; ℤ).

Theorem 4.6. Let n = dim Δ. Then the reduced cohomology H̃^{i}(X_{∞}; ℤ) is 0 for i ≠ n − 1 and is a free abelian group for i = n − 1.

Proof. For 1 < m < N let L′_{m} be the complement of L_{m} in 𝔠̄_{0}. Then L′_{m} is the union of the closures of the faces (𝔠_{0})_{{s}} of codimension one for s ∈ S − I_{m}. Since I_{m} ≠ ∅ and I_{m} ≠ S, the subspace L′_{m} is contractible. The exact cohomology sequence

⋯ → H̃^{i−1}(L′_{m}; ℤ) → H_{c}^{i}(L_{m}; ℤ) → H̃^{i}(𝔠̄_{0}; ℤ) → H̃^{i}(L′_{m}; ℤ) → ⋯

implies that H_{c}^{i}(L_{m}; ℤ) = 0 for all i. By Proposition 4.5 and the Künneth Theorem it follows that H_{c}^{∗}(X_{m} − X_{m−1}; ℤ) = 0.

The space X_{1} = 𝔠̄_{0} is contractible, so its reduced cohomology is zero. Applying the long exact cohomology sequence inductively to the pairs (X_{m}, X_{m−1}), we conclude that H^{∗}(X_{m}; ℤ) = 0 for m < N and that

H̃^{∗}(X_{∞}; ℤ) ≅ H_{c}^{∗}(X_{∞} − X_{N−1}; ℤ).

By Proposition 4.5 there is a homeomorphism between X_{∞} − X_{N−1} and ℭ_{N} × L_{N} = ℭ_{N} × 𝔠_{0}. The space 𝔠_{0} is homeomorphic to ℝ^{n−1}, so

H_{c}^{i}(𝔠_{0}; ℤ) = \begin{cases}ℤ & if i = n − 1 \\ 0 & otherwise. \end{cases}

COHOMOLOGY OF BUILDINGS AND FINITENESS PROPERTIES OF Ã_{n}-GROUPS 15

Thus the Künneth Theorem implies that

H_{c}^{i}(X_{∞} − X_{N−1}; ℤ) ≅ H_{c}^{i−n+1}(ℭ_{N}; ℤ).

Because ℭ_{N} is totally disconnected, H_{c}^{i}(ℭ_{N}; ℤ) = 0 for i > 0 so that H̃^{i}(X_{∞}; ℤ) = 0 for i ≠ n−1. Moreover, H_{c}^{0}(ℭ_{N}; ℤ) is a free abelian group because ℭ_{N} is a locally compact Hausdorff space (see Section 4.4 of [Ma]). Thus H̃^{n−1}(X_{∞}; ℤ) is free abelian, and this completes the proof.

Corollary 4.7. Let n = dim Δ. Then the cohomology H_{c}^{i}(Δ; ℤ) is 0 for i ≠ n and is a free abelian group for i = n.

Proof. The space Δ is homeomorphic to X, which is an open subspace of the contractible space X̄. Since X_{∞} = X̄ − X is closed and X̄ is compact, the exact cohomology sequence

⋯ → H̃^{i−1}(X̄; ℤ) → H̃^{i−1}(X_{∞}; ℤ) → H_{c}^{i}(X; ℤ) → H̃^{i}(X̄; ℤ) → ⋯

implies that H_{c}^{i}(Δ; ℤ) ≅ H_{c}^{i}(X; ℤ) ≅ H̃^{i−1}(X_{∞}; ℤ) for all i. The desired result now follows from Theorem 4.6.

5. Finiteness properties of Ã_{n}-groups

In this section we prove that finitely presented Ã_{n}-groups satisfy a number of other finiteness properties. In particular, these groups are virtually torsion-free and have finite virtual cohomological dimension. The results of the previous section imply that they are virtual duality groups, and we end by showing that they have only finitely many conjugacy classes of finite subgroups.

A group Γ is an Ã_{n}-group if it acts simply transitively on the vertices of a thick Ã_{n}-building in a type-rotating manner. By the results of [CMSZ1] and [CMSZ2] for n = 2 and of [C] for n > 2 it is known that Γ has a simple presentation in terms of a projective geometry. In particular, suppose that Π is an n-dimensional projective geometry, and let Π_{i} denote the set of elements of dimension i in Π. Suppose that λ : Π → Π is an involution satisfying λ(Π_{i}) = Π_{n+1−i} for 1 ≤ i ≤ n. A set T of ordered triples of elements in Π is called an Ã_{n} triangle presentation compatible with λ if it satisfies the following conditions:

(1) Let u, v ∈ Π. Then there is a w ∈ Π such that (u, v, w) ∈ T if and only if λ(u) and v are distinct and non-incident.

(2) If (u, v, w) ∈ T , then (v, w, u) ∈ T .

(3) If (u, v, w) ∈ T and (u, v, w′) ∈ T , then w = w′.

(4) If (u, v, w) ∈ T , then (λ(w), λ(v), λ(u)) ∈ T .

(5) If (u, v, w) ∈ T , then dim u + dim v + dim w ≡ 0 (mod n + 1).

16 JACQUI RAMAGGE AND WAYNE W. WHEELER

(6) For i = 1, 2 set T^{i} = {(u, v, w) ∈ T | dim u + dim v + dim w = i(n + 1)}. If (u, v, w) ∈ T^{2} and (u, v′, w′) ∈ T^{1}, then there exists an element x ∈ Π such that (λ(v), v′, x) ∈ T^{1} and (w, λ(w′), x) ∈ T^{2}.

If T is an Ã_{n} triangle presentation, then there is an associated group Γ_{T} generated by the symbols a_{u} for u ∈ Π and satisfying the relations a_{u}a_{λ(u)} = 1 for u ∈ Π and a_{u}a_{v}a_{w} = 1 for (u, v, w) ∈ T . The results of [CMSZ1], [CMSZ2], and [C] show that the Cayley graph of Γ_{T} with respect to this presentation is the 1-skeleton of a thick Ã_{n}-building Δ_{T}. In particular, Γ_{T} acts simply transitively on the vertices of Δ_{T} in a type-rotating manner, so it is an Ã_{n}-group. Conversely, every Ã_{n}-group is isomorphic to Γ_{T} for some Ã_{n} triangle presentation T .

For the remainder of this section we assume that T is an Ã_{n} triangle presentation as above. Set Γ = Γ_{T} and Δ = Δ_{T}. Since dim u + dim v + dim w ≡ 0 (mod n + 1) for all (u, v, w) ∈ T and dim λ(u) = n + 1 − dim u for all u ∈ Π, the map τ : Γ → ℤ_{n+1} defined on generators of Γ by τ(a_{u}) = dim u extends to a group epimorphism. We begin by studying the properties of the subgroup K = Ker τ and of the stabilizer Γ_{σ} of a simplex σ in Δ.

Proposition 5.1. Let σ be any simplex in Δ. If γ ∈ Γ_{σ} − {1}, then γ is conjugate to a generator of Γ of order dividing n + 1. Moreover, K_{σ} = 1.

Proof. Identify the vertices of Δ with the elements of Γ and the 1-skeleton with the Cayley graph. Because γ ≠ 1, no vertex of σ is fixed by γ. In particular, if v is a vertex of σ, then γv ≠ v is also a vertex of σ because γ ∈ Γ_{σ}. Thus γv = va_{u} for some generator a_{u} of Γ. Since γ permutes the vertices of σ, we have γ^{m}v = v for some m > 0. Thus γ^{m} = 1, and a_{u} = v^{−1}γv is a generator of finite order.

If γ′ ∈ K_{σ} − {1}, then γ′ is conjugate to some generator a_{u′}. But then

dim u′ = τ(a_{u′}) ≡ τ(γ′) ≡ 0 (mod n + 1),

and we have a contradiction. Hence K_{σ} = 1. Since K is normal in Γ and |Γ : K| = n + 1, it follows that γ^{n+1} ∈ K ∩ Γ_{σ} = 1. Thus γ is conjugate to a generator of order dividing n + 1, and this completes the proof.

Proposition 5.2. Let H be a subgroup of Γ. Then H is finite if and only if H stabilizes some simplex σ of Δ. In particular, Γ_{σ} is cyclic of order dividing n + 1 for every simplex σ of Δ.

Proof. Assume that H is finite, and let x be any point in Δ. Then the orbit Hx = {h·x | h ∈ H} is bounded. By the Bruhat–Tits Fixed Point Theorem there is a point y ∈ Δ that is fixed by H. Let σ be the unique simplex of minimum dimension containing y. Then H stabilizes σ.

COHOMOLOGY OF BUILDINGS AND FINITENESS PROPERTIES OF Ã_{n}-GROUPS 17

To prove the remainder of the proposition, it is only necessary to observe that if σ is any simplex of Δ, then Γ_{σ} ≅ Γ_{σ}K/K is a subgroup of Γ/K ≅ ℤ_{n+1}. Thus Γ_{σ} is cyclic of order dividing n + 1, as claimed.

Propositions 5.1 and 5.2 show that any element γ ∈ Γ − {1} of finite order is conjugate to a generator of order dividing n+1. The proof given here was previously known to D. Cartwright, who has also found a purely algebraic proof for Ã_{2}-groups.

The remaining results are proven under the assumption that the projective geometry Π is finite. This condition implies that Γ is finitely generated and that the building Δ is locally finite so that the results of the previous sections are applicable.

Lemma 5.3. Assume that the projective geometry Π is finite. Then Δ has finitely many simplices modulo K.

Proof. Let v be any 0-simplex in Δ. Since Π is finite, there are only finitely many simplices σ_{1}, …, σ_{s} having v as a vertex. Let σ′ be any simplex of Δ, and let v′ be a vertex of σ′. Then there is an element γ ∈ Γ with γv′ = v, so γσ′ = σ_{i} for some i with 1 ≤ i ≤ s. Thus Δ has only finitely many simplices modulo Γ. Since |Γ : K| = n + 1, it follows that Δ also has finitely many simplices modulo K, as desired.

A group K is said to be of type FP if ℤ admits a projective resolution over ℤK that has finite length and consists of finitely generated modules. If there is a largest integer d ≥ 0 such that H^{d}(K, M) ≠ 0 for some ℤK-module M, then K is said to have cohomological dimension cd K = d.

Proposition 5.4. Assume that the projective geometry Π is finite. Then K ⊆ Γ is a torsionfree subgroup of type FP, and cd K ≤ n.

Proof. The building Δ is contractible, and it is a free K-complex by Proposition 5.1. Thus the augmented cellular chain complex C_{∗}(Δ) of Δ is a free resolution of ℤ over ℤK by Proposition I.4.1 of [B1]. Since Δ has finitely many simplices modulo K, it follows that C_{m}(Δ) is a finitely generated ℤK-module for all m. Hence K is of type FP. Moreover, cd K ≤ n because C_{m}(Δ) = 0 for m > n, and Corollary VIII.2.5 of [B1] implies that K is torsion-free.

Theorem 5.5. Assume that the projective geometry Π is finite. Then Γ is a virtual duality group.

Proof. The building Δ is a contractible K-complex with only finitely many cells modulo K, and K_{σ} = 1 for every simplex σ. These conditions imply that H^{∗}(K, ℤK) ≅ H_{c}^{∗}(Δ; ℤ) (for example, see [B1], p. 209, Exercise 4). Moreover, H_{c}^{n}(Δ; ℤ) is torsion-free, and H_{c}^{i}(Δ; ℤ) = 0 for i ≠ n. Since K is of type FP, Theorem VIII.10.1 of [B1] implies that it is a duality group, and the desired result follows.

In many cases it is possible to give a shorter proof of Theorem 5.5. For example, suppose that n > 2. Then Γ is an S-arithmetic group by Corollary 10.25 of [R] and Margulis’ Arithmeticity Theorem (Theorem 10.1.12 of [Z]). Hence K is torsion-free and S-arithmetic, so it is a duality group by [BS].

Now suppose that n = 2 and that the projective plane Π is desarguesian. Then Γ has Kazhdan’s Property (T) by [CMS], and Corollary III.2.15 of [M] implies that K also has Property (T) because |Γ : K| is finite. Thus K does not decompose as a free product by Theorem III.3.9 of [M]. Since cd K ≤ 2, it follows that K is a duality group (see [B1], p. 185, Examples 1 and 2, and p. 223, Remark 5). The authors are grateful to G. Robertson for pointing out this argument.

Proposition 5.6. Assume that the projective geometry Π is finite. Then Γ has only finitely many conjugacy classes of finite subgroups.

Proof. If H is a finite subgroup of Γ, then Proposition 5.2 implies that H is cyclic and stabilizes some simplex of Δ. Let h be a generator of H. Then h is conjugate to a generator a_{u} ∈ Γ of order dividing n + 1 by Proposition 5.1. Since Π is finite, the number of subgroups ⟨a_{u}⟩ is also finite, and the result follows.

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[B2] K. S. Brown, Buildings, Springer-Verlag, New York, 1989.

[B3] K. S. Brown, Five lectures on buildings; in Group Theory from a Geometrical Viewpoint, E. Ghys, A. Haefliger, and A. Verjovsky, eds., World Sci. Publishing, River Edge, NJ (1991) 254–295.

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[R] M. Ronan, Lectures on Buildings, Academic Press, New York, 1989.

[RR] J. Ramagge and G. Robertson, Triangle buildings and actions of type III_{1/q^{2}}, J. Funct. Anal. 140 (1996) 472–504.

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Mathematics Department, University of Newcastle, NSW 2308, Australia

   E-mail address: jacqui@maths.newcastle.edu.au

Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester LE1 7RH, England

   E-mail address: w.wheeler@mcs.le.ac.uk