Coxeter Complexes
3.2 Local Properties of Coxeter Complexes
Remark 3.18.We have included this corollary only to force the reader to learn the terminology associated with chamber systems, especially the concept ofresidue. But the statement of the corollary is in fact a complete tautology in view of the definition ofΣin terms of standard cosets. Indeed, if one identifies chambers with elements of W, then it is immediate from the definitions that the residues are the standard cosets.
Proposition 3.16 has a simple interpretation in terms of Coxeter matrices.
Recall, first, that the Coxeter system (W, S) is determined by its Coxeter matrix M =
m(s, t)
s,t∈S. So we may think of Σ as a simplicial complex associated to M. Next, note that the rows and columns of M are indexed byS, which is also the set of types of the vertices ofΣ. What the proposition says, then, is that lkA is the Coxeter complex associated to the matrix MJ obtained fromM by selecting the rows and columns belonging to the cotypeJ ofA.
This becomes even easier to use if we translate it into the language of Coxeter diagrams. Recall that the diagram of (W, S) is a graphD with labels on some edges. There is one vertex for each s ∈ S, with s joined to t if m(s, t)≥3, and with a label over that edge ifm(s, t)≥4. The passage from M toMJabove, and hence the passage fromΣto lkA, corresponds to passing to theinduced subdiagram DJ with vertex setJ equal to the cotype ofA. In other words, we retain the vertices in the cotype (and all edges between them).
Equivalently, we delete all vertices inτ(A) (and all edges touching them).
Consider, for example, the groupW = PGL2(Z) studied in Section 2.2.3.
Its diagram is
∞ .
The Coxeter complexΣ has rank 3 (dimension 2), so there are three types of vertices. Let’s compute the link of each type of vertex.
According to the recipe above, we must delete one vertex at a time from the Coxeter diagram of W. This yields the Coxeter diagrams of the dihedral groupsD2m, wherem=∞, 2, and 3, respectively. Now it is easy to figure out what the Coxeter complex associated to D2mlooks like, and in fact, we have already seen it in Chapters 1 and 2. Namely, it is a 2m-gon; in other words, it is a triangulated circle with 2medges ifm <∞, and it is a triangulated line ifm=∞. So our three links in this example are a line, a quadrilateral, and a hexagon.
Exercise 3.19.Look at Figure 2.5. Can you find the three types of links in the picture?
This example illustrates a general principle, valid for all Coxeter com- plexes: The link of a codimension-2 simplex of cotype {s, t} (withs=t) is a 2m-gon, wherem=m(s, t). This fact yields a geometric interpretation of the Coxeter matrix M:
3.2 Local Properties of Coxeter Complexes 121 Corollary 3.20.The Coxeter matrix M of (W, S) can be recovered from Σ as follows: For any s, t ∈ S with s = t, m(s, t) is the unique number m (2≤m≤ ∞)such that the link of every simplex of cotype{s, t}is a 2m-gon.
This shows, in particular, that the Coxeter groupW is determined up to isomorphism byΣ. We will see this again in the next section, from a different point of view.
Remark 3.21.Note that a 2m-gon has diameterm, where thediameter of a chamber complex is the supremum of the gallery distances between its cham- bers. So we can also write the geometric interpretation ofM as
m(s, t) = diam(lkA),
whereAhas cotype{s, t}as above. The result in this form is valid even when s=t. [In this case the link has exactly two chambers, which are adjacent, so the diameter is indeed 1 =m(s, s).]
We can use this corollary, together with Tits’s solution to the word problem for Coxeter groups, to give a simple answer to a question that might seem, a priori, to be very difficult: How can one describe the totality of minimal galleries connecting two given chambers? This is easy in the 1-dimensional case, whereΣis a 2m-gon: Minimal galleries are unique unlessm <∞and the two given chambersC1 andC2 are at maximum distancemfrom each other, i.e., they are opposite. In this case there are exactly two minimal galleries connectingC1to C2.
Translating this result to the link of a simplexA of codimension 2 in an arbitrary Coxeter complex, we obtain a similar description of the minimal galleries in the subposet Σ≥A. Visualize, for example, the case that Σ is 2-dimensional andAis a vertexvwhose link is finite. Then for somem <∞, Σ≥A contains 2mchambers that form a solid 2m-gon centered atv. The only nonuniqueness of minimal galleries in this subposet arises from the fact that there are two ways of going around the 2m-gon to get from a given chamber to the opposite chamber.
Since galleries correspond to words, we can use the solution to the word problem (Section 2.3.3) to analyze the general case. The answer, roughly, is that the nonuniqueness of minimal galleries in a Coxeter complex can be explained entirely in terms of the obvious nonuniqueness that occurs in links of codimension-2 simplices. To state this precisely, we need some terminology.
Definition 3.22.If Γ:C0, . . . , Cd is a gallery, then the type of Γ is the se- quence s:= (s1, . . . , sd) such thatCi−1 issi-adjacent toCi fori= 1, . . . , d.
(This notion of “type of a gallery” makes sense in any colorable chamber complex; we will use it again later.)
Suppose Γ has a subgallery of type (s, t, s, t, . . .) and of length m = m(s, t) < ∞, where s = t. Then this subgallery lies in Σ≥A for some codimension-2 simplexAof cotype{s, t}, and we may replace it by the other minimal gallery inΣ≥Awith the same extremities. This produces a new gallery Γ fromC0to Cd.
Definition 3.23.The gallery Γ is said to be obtained from Γ by an ele- mentary homotopy. Two galleries are said to behomotopic if there is a finite sequence of elementary homotopies transforming one to the other.
Figure 3.2 shows an elementary homotopy from a gallery of type (u, t, s, t, u) to one of type (u, s, t, s, u).
u u
t s u t
u t s
t C
D u
s
Fig. 3.2.An elementary homotopy.
The following result is now immediate from our earlier observations:
Proposition 3.24.Any two minimal galleries with the same extremities are
homotopic.
Remark 3.25.We have framed our discussion in terms of links. We could equally well have used the language of residues. Indeed, ifAis a codimension-2 simplex of cotype{s, t}as above, then the set of chambers inΣ≥Ais a residue of type{s, t}. So our elementary homotopies all take place in rank-2 residues.
Here therank of a residue is the cardinality of its type, which is the same as the codimension of the corresponding simplexA.
Exercise 3.26.Give a method for using homotopies to decide whether a given gallery is minimal and if not, to obtain a minimal gallery from it.
Finally, we can use our calculation of links to answer another natural question, at least to readers who are familiar with combinatorial topology:
3.2 Local Properties of Coxeter Complexes 123 When is Σ a manifold? This question arises naturally because triangulated manifolds (without boundary) are the canonical examples of thin chamber complexes (Example A.9). We already know the answer ifW is finite: In this caseΣis a sphere (see the remarks following Definition 3.2); in particular,Σ is a manifold.
What happens if W and Σ are infinite? There is an obvious necessary condition. Namely, manifolds are locally compact, hence locally finite, i.e., every nonempty simplexAis a face of only finitely many chambers. In other words, the link ofAmust be finite. Conversely, if the link of every nonempty simplex is finite, then it is in fact a sphere (since it is a finite Coxeter complex).
We leave it as an exercise for the interested reader to deduce thatΣ is then a manifold. This proves the following:
Corollary 3.27.The following conditions are equivalent:
(i)Σ is a manifold.
(ii)Σ is locally finite.
(iii)Every proper standard subgroup of W is finite.
For example, the Coxeter complex associated to PGL2(Z) is not a mani- fold. One can see the nonmanifold points in Figures 2.3 and 2.5: They are the cusps.
Exercise 3.28.If (iii) holds andW is infinite, show that (W, S) is irreducible.
Remark 3.29.Condition (iii) is quite restrictive. One can show that it holds only in the following three cases: (a)W is finite; (b)Wis an irreducible Euclid- ean reflection group; (c)W is a hyperbolic reflection group whose fundamental domain is a closed simplex contained entirely in the interior of the hyperbolic space. See Chapter 10 for definitions of the terms used in (b) and (c) and for more information.
Even though we have not yet officially discussed Euclidean reflection groups, most readers probably have some intuition about them. For exam- ple, D∞ is a Euclidean reflection group acting on the line, and the group of Example 3.7 is a Euclidean reflection group acting on the plane. It is natural to wonder why reducible Euclidean reflection groups were excluded in Remark 3.29 (and in Exercise 3.28): Given Euclidean reflection groups W1 andW2 acting on Euclidean spaces E1 and E2, isn’t their product W a Euclidean reflection group acting onE=E1×E2, which is a Euclidean space and hence a manifold? And doesn’tΣtriangulate this manifold? The answer is “yes” to the first question, but “no” to the second. The following exercise explains what happens.
Exercise 3.30.Let (W, S) and (W, S) be Coxeter systems, and let (W, S) be their product (withW :=W×W andS:=S∪S). Show that
Σ(W, S)∼=Σ(W, S)∗Σ(W, S),
where the asterisk denotes the join operation. [Recall that thejoin ∆ of two simplicial complexes ∆ and ∆ with vertex sets V and V has vertex set equal to the disjoint union V V and has one simplex A ∪A for every A ∈ ∆ and A ∈ ∆. From the poset point of view, then, ∆ is simply the Cartesian product of∆ and∆. But its geometric realization|∆|isnot the Cartesian product |∆| × |∆|; in fact, ∆ does not even have the right dimension for this to be true.]
Returning to the question whether a reducible Coxeter group can yield a Coxeter complex that is a manifold, the essential point is that the join of two manifolds that are not spheres is generally not a manifold. [But the join of two spheres is a sphere.]
This discussion suggests that the Coxeter complexΣ(W, S) is not always the “best” geometric model for a Coxeter group W. For example, it would seem more reasonable to use a product of Euclidean spaces rather than a join of Euclidean spaces in the case of a reducible Euclidean reflection group.
The result is a cell complex whose cells are products of simplices rather than simplices. We will return to this circle of ideas in Chapter 12.