Non-positively curved aspects of Artin groups of finite type

Mladen Bestvina^{∗}

November 1998

1 Introduction and Review

A theme in geometric group theory has been to, on the one hand, construct spaces with rich geometry where (given) interesting groups act by isometries, and on the other hand, show that such actions have useful group-theoretic consequences. A particularly successful sort of a geometric structure one considers is a CAT(0) metric. Experience shows that the class of groups that act cocompactly and isometrically on CAT(0) spaces is large and includes many standard groups, while at the same time the groups in this class have a rich structure (see [BH]). For example, the following properties hold in this class:

• There are only finitely many conjugacy classes of finite subgroups.

• Every solvable subgroup is finitely generated and virtually abelian.

• The set of translation lengths of elements of infinite order is bounded away from 0.

In this paper, we focus on the class of Artin groups of finite type (and their quotients by the center). We review the basic definitions and results in the next two sections. Tom Brady [Bra] has shown that Artin groups of finite type with ≤ 3 generators belong to the CAT(0) class. Daan Krammer has proposed a simplicial complex with a metric for each Artin group of finite type, and for small numbers of generators the metric is CAT(0).

It is a result of Ruth Charney [Cha92] that Artin groups of finite type are biautomatic. This fact has a number of group-theoretic consequences; in particular, by a result of Gersten-Short [GS91], the translation length of each element of infinite order is positive, and consequently every nilpotent subgroup is virtually abelian.

In this paper, we will construct a contractible simplicial complex on which a given Artin group of finite type acts, and show that there is a natural “metric” on it, preserved by the group action, that satisfies a property somewhat weaker than CAT(0). The word “metric” is in quotes because it is not symmetric. This structure is, however, sufficient to deduce the same group-theoretic properties as in the case of CAT(0) metrics.

The main tool in this paper is the left greedy normal form for the elements of Artin groups of finite type. The basic properties of this normal form were established in the inspiring paper of Deligne [Del72], and also in Brieskorn-Saito [BS72], both of which build on the work of Garside [Gar69]. Charney [Cha92] showed that this form gives rise to a biautomatic structure. Deligne also constructed a finite Eilenberg-Mac Lane space for each Artin group of finite type.

Aside from establishing the above group-theoretic properties of Artin groups of finite type and their central quotients, we will give a simple proof of Squier’s theorem [Squ94] that these groups are duality groups. Moreover, we show that they are highly connected at infinity. (One way to show that a group is a duality group of dimension d is to argue that the group is d-dimensional and d − 2-connected at infinity.) We also show that every normal abelian subgroup is central. This answers a question of Jim Carlson that provided the impetus to study the geometry of Artin groups of finite type.

∗Support by the National Science Foundation is gratefully acknowledged.

We summarize the geometric features of the complex in the following (see the next section for definitions of S and Δ):

Main Theorem. Let A be an Artin group of finite type and G = A/Δ^{2} the quotient of A by the central element Δ^{2}. Then G acts simplicially on a simplicial complex X = X(G) with the following properties:

• The action is cocompact and transitive on the vertices (section 2.1).

• X is a flag complex (section 2.1).

• X is contractible, (card(S) − 3)-connected at infinity and proper homotopy equivalent to a (card(S) − 1)-complex (Theorem 3.5).

• Each pair of vertices of X is joined by a preferred edge-path, called a geodesic (section 3.1). The collection of geodesics is invariant under the following operations: translation by group elements, subpaths, inverses (Proposition 3.1), and concatenations with nontrivial overlap.

• There is a function d = d_{wd} : X^{(0)} × X^{(0)} → {0, 1, 2, ⋯ } (a “non-symmetric distance function”) satisfying the triangle inequality and d(v, w) = 0 ⟺ v = w (Proposition 3.1). Further, d_{wd} is comparable to the edge-path metric in X (Lemma 3.2).

• If v_{0}, v_{1}, v_{2}, ⋯ v_{n} are the consecutive vertices along a geodesic, then

d(v_{0}, v_{n}) = ∑_{i=1}^{n} d(v_{i−1}, v_{i})

• If the associated Coxeter group W is irreducible and nonabelian, then every geodesic can be extended in both directions to a longer geodesic (Proposition 4.9).

• (Theorem 3.14) For any three vertices a, b, c ∈ X and any vertex p ≠ b, c on the geodesic from

b to c we have

d(a, p) < max{d(a, b), d(a, c)}.

• The set of vertices of each simplex in X has a canonical cyclic ordering which is invariant under the group action and is compatible with the passage to a face (Lemma 4.6).

The dimension of the complex we construct is higher than the expected dimension (i.e. the dimension of the group). It is, however, often the case that one has to increase the dimension in order to obtain better geometric properties, cf. non-uniform lattices, Teichmüller space, Outer Space.

Perhaps the most interesting remaining unresolved group-theoretic questions about Artin groups A of finite type are the following: Question 1: Does A satisfy the Tits Alternative, i.e. if H < A is a subgroup which is not virtually abelian, does H necessarily contain a nonabelian free group? Question 2: Is A virtually poly-free?

The Tits Alternative is not known for CAT(0) groups and it seems unlikely that the techniques of this paper will resolve Question 1. A group G is poly-free if there is a finite sequence

G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ ⋯ ⊃ G_{n} = {1}

such that each G_{i} is normal in G_{i−1} and the quotient G_{i−1}/G_{i} is free. It would seem reasonable to expect that the pure Artin group P A (the kernel of the homomorphism to the associated Coxeter group) is poly-free, with the length of the series above equal to the dimension (i.e. the number of generators), and all successive quotients of finite rank. This was verified by Brieskorn [Bri73] for types A_{n} (this is classical), B_{n}, D_{n}, I_{2}(p), and F_{4}. Of course, the positive answer to Question 2 implies the positive answer to Question 1. Another approach to Question 1 would be to find a faithful linear representation of each Artin group of finite type. This seems rather difficult (if not impossible) even for braid groups.

Questions 1 and 2 can be asked in the setting of Artin groups of infinite type as well. However, more basic questions are still open in that setting; the most striking is whether all (finitely generated) Artin groups admit a finite K(π, 1). Recently, Charney and Davis [CD95b],[CD95a] have made substantial progress toward this question, but the general case remains open.

We end the paper by constructing a space at infinity analogous to the space of projectivized measured geodesic laminations in the case of mapping class groups.

I benefited from discussions about Artin groups with several people. I would like to thank John Luecke and Alan Reid for having the patience to listen to me while I was learning about Artin groups. A joint work in progress with Mark Feighn investigates the structure of axes of elements in more detail, with applications to the dynamics of the action on the space at infinity. Discussions with Warren Dicks have shed much light on the difficulties in Question 2.

1.1 Coxeter groups

We follow the notation of [CD95a]. For proofs of the facts listed below see [Bou81]. Let S be a finite set. A Coxeter matrix is a symmetric function m : S × S → {1, 2, 3, ⋯, ∞} such that m(s, s) = 1 for all s ∈ S and m(s, s′) ≥ 2 for s ≠ s′. The associated Coxeter group is the group W given by the presentation

W =< S|(ss′)^{m(s,s′)} = 1 >

where m = ∞ means no relation. If S′ is a proper subset of S and m′ is the restriction of m to S′ × S′, then there is a natural homomorphism

W′ =< S′|(ss′)^{m′(s,s′)} = 1 >⟶ W

and this homomorphism is injective. The images of such homomorphisms are called special (Coxeter) subgroups of W.

In this paper we will consider only finite Coxeter groups W, and this is what we assume from now on.

There is a canonical faithful orthogonal representation of W on a finite-dimensional vector space V = V_{W} (of dimension dim V = card(S)) such that:

• Each generator s ∈ S acts as a reflection, i.e. it fixes a codimension 1 subspace of V. These subspaces and their W-translates are called walls.

• The closures in V of the complementary components of the union of the walls are the chambers. They are simplicial cones, and W acts on the set of chambers simply transitively.

• Special subgroups and their conjugates are precisely the stabilizers of nonzero points in V.

• There is a chamber Q, called the fundamental chamber, that intersects the fixed sets of each s ∈ S in top-dimensional faces. When W is irreducible, Q and −Q are the only such chambers.

• The longest element is the unique element Δ ∈ W that takes Q to −Q (and vice versa). It has order 2, and conjugation by Δ induces an involution of S.

Two chambers Q′ and Q′′ are adjacent if Q′ ∩Q′′ is contained in a wall and has nonempty interior in the wall. This wall is then the unique wall that separates Q′ from Q′′, and we say that it abuts Q′ and Q′′. A sequence Q_{0}, Q_{1}, ⋯, Q_{n} of chambers is a gallery G of length n (from Q_{0} to Q_{n}) if Q_{i} and Q_{i+1} are adjacent for i = 0, 1, 2, ⋯, n−1. Any path in U which is transverse to the collection of the walls determines a gallery and every gallery arises in this way. Any two chambers are connected by a gallery. The distance between two chambers A, B is the length of a shortest gallery connecting them; equivalently, it is the number of walls separating A and B. A gallery is geodesic if its length is equal to the distance between the initial and the terminal chamber. Equivalently, a gallery is geodesic if it crosses each wall at most once (we say that a gallery crosses a wall if any associated path does, i.e. if the wall separates a pair of consecutive chambers). For example, straight line segments transverse to the collection of walls determine geodesic galleries.

A Coxeter group W is irreducible if there is no nontrivial partition S = S_{1} ⊔S_{2} with m(s_{1}, s_{2}) = 2 whenever s_{1} ∈ S_{1} and s_{2} ∈ S_{2}.

1.2 Artin groups

If S is a finite set and m : S × S → {2, 3, ⋯, ∞} a Coxeter matrix, the associated Artin group is defined to be the group

A =< S|prod(s, s′, m(s, s′)) = prod(s′, s, m(s′, s)) >

where

prod(s, s′, m) = \underbrace{ss′ss′ ⋯}_{m times}

and again m = ∞ means no relation.

There is a natural homomorphism π : A → W, π(s) = s. The kernel is the pure Artin group P. If S′ is a subset of S, let A′ be the Artin group associated with the restriction of m to S′ × S′. We then have a natural homomorphism A′ → A, s′ ↦ s′. This homomorphism is injective.

By A^{+} we denote the submonoid of A generated by S, i.e. the monoid of positive words. As shown by Deligne and Brieskorn-Saito, if two positive words represent the same element of A, then they can be transformed to each other by repeated substitutions given by the defining relations. A positive word s_{i_{1}}s_{i_{2}} ⋯ s_{i_{k}} can be geometrically thought of as a gallery of length k starting at Q: Q, s_{i_{1}}(Q), s_{i_{1}}s_{i_{2}}(Q), ⋯, s_{i_{1}}s_{i_{2}} ⋯ s_{i_{k}}(Q). We will also consider any W-translate of this gallery as being associated with the same positive word. Multiplication in A^{+} corresponds to the concatenation of galleries (after a possible W-translation so that the last chamber of the first gallery coincides with the first chamber of the second gallery).

1.3 Normal form in A^{+} and A

Following Deligne, for x, y ∈ A^{+} we write x < y if there is z ∈ A^{+} such that y = xz. We say that A ∈ A^{+} is an atom if its word-length in A^{+} is equal to the word-length of π(A) ∈ W, or equivalently if a gallery associated to A is geodesic. The homomorphism π : A → W induces a bijection between the set of atoms and W. We will often identify an atom A with the associated chamber π(A)(Q). The atom that corresponds to the longest element Δ is also denoted Δ. We have 1 < A < Δ for all atoms A. Δ^{2} is central in A and conjugation by Δ induces an involution of A that extends the involution S → S mentioned above. This involution will be denoted by x ↦ \overline{x}. It restricts to involutions of A^{+} and of the set of atoms.

The following two propositions are proved in Deligne’s paper [Del72].

Proposition 1.1. Suppose that Ω is a nonempty finite subset of A^{+} such that

1. x ∈ Ω, x′ < x implies x′ ∈ Ω, 2. if σ, τ ∈ S are two generators, x ∈ A^{+}, xσ, xτ ∈ Ω, then

xprod(σ, τ, m(σ, τ)) ∈ Ω.

Then there is a unique y ∈ A^{+} such that

Ω = {x ∈ A^{+}|x < y}.

Further, for any y the set Ω = {x ∈ A^{+}|x < y} satisfies 1 and 2.

In particular, for x_{1}, x_{2} ∈ A^{+} we can define x_{1} ∧ x_{2} ∈ A^{+} as the largest element z ∈ A^{+} such that z < x_{1} and z < x_{2}. The existence of the largest such element follows by applying Proposition 1.1 to the set Ω = {z|z < x_{1}, z < x_{2}}.

There is a function reverse : A^{+} → A^{+} given by reverse(s_{i_{1}}s_{i_{2}} ⋯ s_{i_{k}}) = s_{i_{k}} ⋯ s_{i_{2}}s_{i_{1}}. If x ∈ A^{+} is represented by a gallery Q_{0}, Q_{1}, ⋯, Q_{k−1}, Q_{k}, then reverse(x) is represented by the gallery Q_{k}, Q_{k−1}, ⋯, Q_{1}, Q_{0}.

For every atom A there is an atom A^{∗} such that AA^{∗} = Δ (and there is also an atom ^{∗}A such that ^{∗}AA = Δ). Every x ∈ A^{+} can be represented as the product of atoms (or even generators), say x = A_{1}A_{2} ⋯ A_{k}. Then we have

xA_{k}^{∗}\overline{A}_{k−1}^{∗}A_{k−2}^{∗} ⋯ = Δ^{k}

so that x < Δ^{k}. Further, if x, y < Δ^{k}, and if x̃, ỹ ∈ A^{+} are such that xx̃ = Δ^{k} and yỹ = Δ^{k}, then x < y ⟺ reverse(ỹ) < reverse(x̃) and we deduce (from the existence of x̃ ∧ ỹ) that for any x, y ∈ A^{+} there is a unique x ∨ y ∈ A^{+} such that

• x < x ∨ y,

• y < x ∨ y, and

• z < x, z < y implies z < x ∨ y.

For every x ∈ A there is k ≥ 0 such that Δ^{k}x ∈ A^{+}. This reduces the understanding of A to the understanding of A^{+}. There is no such reduction when the associated Coxeter group is infinite.

Proposition 1.2. For every g ∈ A^{+} there is a unique atom α(g) such that

1. α(g) < g, and

2. if A is an atom with A < g, then A < α(g).

Furthermore, α(xy) = α(xα(y)) for all x, y ∈ A^{+}.

The left greedy normal form of g ∈ A^{+} is the representation of g as the product of nontrivial atoms

g = A_{1}A_{2} ⋯ A_{k}

such that A_{i} = α(A_{i}A_{i+1} ⋯ A_{k}) for all 1 ≤ i ≤ k. To emphasize that the above product of atoms is the normal form of g, we will write g = A_{1} · A_{2} · · · · · A_{k}. Clearly, the normal form for g is unique. A product A_{1}A_{2} ⋯ A_{k} is a normal form iff each subproduct of the form A_{i}A_{i+1} is a normal form. If A_{1} · A_{2} · · · · · A_{k} is a normal form, then so is \overline{A}_{1} · \overline{A}_{2} · · · · · \overline{A}_{k}.

Δ’s appear at the beginning of the normal form, i.e. if A_{1} · A_{2} · · · · · A_{k} is a normal form, then there is j such that A_{1} = A_{2} = ⋯ = A_{j} = Δ and A_{i} ≠ Δ for i > j.

Geometric interpretation of the normal form is given by

Proposition 1.3. Let A and B be nontrivial atoms, with A represented by a geodesic gallery from Q_{1} to Q_{2} and B by a geodesic gallery from Q_{2} to Q_{3}. Then A · B is a normal form iff no wall that abuts Q_{2} has Q_{1} and Q_{2} on one side and Q_{3} on the other side.

There is the homomorphism length : A → ℤ that sends each s ∈ S to 1 ∈ ℤ. This homomorphism restricts to the word-length on A^{+}.

We will need a straightforward generalization of Proposition 1.2.

Proposition 1.4. Suppose x ∈ A^{+} and y_{1} · y_{2} · · · · · y_{l} is a normal form in A^{+}. Suppose the product

x(y_{1} · y_{2} · · · · · y_{l})

has normal form

z_{1} · z_{2} · · · · · z_{m}.

Then for each i ≤ l the normal form of the product

x(y_{1}y_{2} ⋯ y_{i})

begins with

z_{1} · z_{2} · · · · · z_{i} ⋯

Proof. For i = 1 this follows from Proposition 1.2. We argue by induction on i. It suffices to show that z_{1}z_{2} ⋯ z_{i} < xy_{1}y_{2} ⋯ y_{i}. By Proposition 1.2 we have z_{1} < xy_{1} so we can write xy_{1} = z_{1}w for some w ∈ A^{+}. Then xy_{1}y_{2} ⋯ y_{l} = z_{1}z_{2} ⋯ z_{m}, after cancelling on the left, implies that wy_{2} ⋯ y_{l} = z_{2} ⋯ z_{m} and the right-hand side is visibly a normal form. By induction it follows that z_{2} ⋯ z_{i} < wy_{2} ⋯ y_{i}. Now multiply on the left by z_{1} to obtain z_{1}z_{2} ⋯ z_{i} < xy_{1}y_{2} ⋯ y_{i}.

2 G and its complex X(G)

2.1 Definition of X(G)

It is more convenient to study the group G = A/Δ^{2}. The groups G × ℤ and A are commensurable. Indeed, the homomorphism A → G × ℤ which is natural projection in the first coordinate and sends each generator to 1 ∈ ℤ in the second coordinate (i.e. it is the length homomorphism in the second coordinate) is a monomorphism onto a finite index subgroup. Moreover, ℤ =< Δ^{2} >↪ A has a splitting with values in \frac{1}{length(Δ^{2})}ℤ (given by length divided by the length of Δ^{2}).

The goal of this section is to describe a contractible simplicial complex X = X(G) on which the group G = A/ < Δ^{2} > acts cocompactly and with finite point stabilizers. The vertex set is the coset space

V = A/ < Δ >= {g < Δ > |g ∈ A}.

The group A acts naturally on the left, and the central element Δ^{2} acts trivially, so G = A/ < Δ^{2} > acts on V. The action is transitive, and the stabilizer of the point < Δ >∈ V is < Δ >≅ ℤ/2.

We call the elements of V the vertices. Each vertex has a representative g ∈ A^{+} (obtained by multiplying an arbitrary representative by a high power of Δ^{2}). Let g = Δ^{m} · B_{1} · B_{2} · · · · · B_{k} be the normal form for g with B_{i} ≠ Δ for all i (as remarked above, if the normal form of g involves any Δ’s, they appear at the beginning). Now shift all Δ’s to the end to obtain a representative in A^{+} of the same coset whose normal form does not have any Δ’s. If m is even, this amounts to replacing g by B_{1} · B_{2} · · · · · B_{k}, and if m is odd g is replaced by \overline{B}_{1} · \overline{B}_{2} · · · · · \overline{B}_{k}. Such a coset representative is unique, and we will frequently identify V with the set of elements of A^{+} whose normal form has no Δ’s. The identity element of A^{+} (whose normal form is empty) is viewed as the basepoint, denoted ∗.

The atomnorm of a vertex v, denoted |v|, is the number of atoms in the normal form of the special representative, i.e. the number of atoms not counting Δ’s in the normal form of any positive representative. Left translation by Δ sends a special representative B_{1} · B_{2} · · · · · B_{m} to a special representative \overline{B}_{1} · \overline{B}_{2} · · · · · \overline{B}_{m} so this involution is atomnorm preserving. It follows that there is a unique left invariant function d_{at} : V × V → ℤ_{+} = {0, 1, 2, ⋯ } (i.e. d_{at}(g(v), g(w)) = d_{at}(v, w) for all g ∈ G and all v, w ∈ V) such that d_{at}(∗, v) is the atomnorm of v. We call d_{at} the atomdistance on V.

In order to prove that d_{at} is symmetric and satisfies the triangle inequality, we need the following lemma.

Lemma 2.1. If g ∈ A^{+} is the product of k atoms, then the normal form of g has ≤ k atoms.

Proof. Induction on k starting with k = 1 when it’s clear. For k = 2 the statement follows from the definition of normal form and the fact that a subgallery of a geodesic gallery is a geodesic gallery. Say g = B_{1}B_{2} ⋯ B_{k}. Inductively, the normal form for B_{2} ⋯ B_{k} is C_{2} · C_{3} · · · · · C_{l} with l ≤ k. If B_{1}C_{2} is an atom, then the normal form for g is (B_{1}C_{2}) · C_{3} · · · · · C_{l} and has length l − 1 < k. If B_{1}C_{2} is not an atom, then the normal form for B_{1}C_{2} is X · Y (say) and the normal form for g is X followed by the normal form for Y C_{3} ⋯ C_{l}. The latter has length < l by induction, so the statement is proved.

Proposition 2.2. d_{at} : V × V → ℤ_{+} is a distance function.

Proof. The triangle inequality follows from Lemma 2.1. We argue that d_{at} is symmetric. By left invariance, it suffices to show that d_{at}(∗, v) = d_{at}(v, ∗) for all v ∈ V. Let g be a group element such that g(v) = ∗; we need to argue that the norm of v equals the norm of g(∗) (which can be viewed as an “inverse” of v; there are two inverses – obtained from each other by applying the “bar” involution). Say v = B_{1} · B_{2} · · · · · B_{k} is the normal form (without Δ’s). Let C_{i} be the unique atom such that C_{i}B_{i} = Δ. Thus ΔC_{i} ∈ G is the inverse of B_{i} and g can be taken to be ΔC_{k}ΔC_{k−1} ⋯ ΔC_{1}. Thus g(∗) is represented by \overline{C}_{k}C_{k−1}\overline{C}_{k−2} ⋯, a product of k atoms. By Lemma 2.1, the norm of g(∗) is ≤ k. Applying the same argument with the roles of v and g(v) reversed, we see that the norms of v and g(v) are equal.

Definition 2.3. X(G) is the simplicial complex whose vertex set is V and a collection v_{0}, v_{1}, ⋯, v_{m} ∈ V spans a simplex iff d_{at}(v_{i}, v_{j}) = 1 for all i ≠ j.

Thus X(G) is a flag complex, i.e. if each pair in a set of vertices bounds an edge in X(G) then this set spans a simplex. The group G acts on X(G) simplicially and cocompactly. Note that the distance d_{at} on V is the edge-path distance between the vertices of X(G).

Examples 2.4. We can explicitly see X(G) in simple cases. When A = ℤ × ℤ, X(G) is the line. When A is ℤ×ℤ×ℤ, X is the plane (triangulated in the usual (3, 3, 3)-fashion), and more generally, when A = ℤ^{m}, then X(G) is Euclidean (m − 1)-space. When A = B_{3} is the braid group on 3 strands, X(G) is the union of triangles glued to each other along their vertices so that the spine is the trivalent tree.

X(G) for G = A/Δ^{2}, A =< a, b|aba = bab >

2.2 Related complexes

One can build a similar complex X′(G) with the vertex set G = A/Δ^{2}. Each vertex has a special representative whose normal form either has no Δ’s, or has a single Δ, and in the latter case we agree to push this Δ to the last slot. There is an oriented edge from a vertex v to a vertex w if the special representative of w is obtained from the special representative of v by rightmultiplying by an atom. The simplicial complex X′(G) is defined to be the flag complex determined by the resulting graph, i.e. a collection of vertices spans a simplex provided all pairs span an edge. The natural quotient map G → G/ < Δ > extends to a simplicial map p : X′(G) → X(G). The preimage of the basepoint ∗ ∈ X(G) is the edge [∗, Δ] ⊂ X′(G). More generally, the preimage of a simplex σ is the (triangulated) prism σ × [0, 1]. There is a global homeomorphism X′(G) ≅ X(G) × [0, 1]. In the first coordinate this homeomorphism is given by p and in the second it is the simplicial map that sends the vertices whose special representative has no Δ’s to 0 and the vertices whose special representative has one Δ to 1.

Analogously, there is a complex X(A) with vertex set A and similarly defined simplicial structure: edges are drawn from ∗ to the atoms and extended equivariantly, and then the higher-dimensional simplices are filled in. The quotient map extends to the natural simplicial map q : X(A) → X(G) and there is a global homeomorphism X(A) ≅ X(G) × ℝ in the first coordinate given by q and in the second it is the map that is linear on each simplex and sends a vertex g ∈ A to length(g) ∈ ℤ (recall that length : A → ℤ is the homomorphism that sends each s ∈ S to 1 ∈ ℤ). The fact that Δ^{2} virtually splits is transparent in this model.

We end this discussion by describing the Deligne complex and showing that it is homotopy equivalent to X(A) (and thus recovering from Theorem 3.5 Deligne’s theorem [Del72] that the Deligne complex is contractible). The Coxeter sphere can be described as (the geometric realization of) the poset of cosets wW_{F} where w ∈ W and W_{F} is the special Coxeter subgroup of W generated by F ⊂ S, as F runs over all proper subsets (including ∅) of S. The partial order is given by inclusion. Here W_{F} corresponds to the barycenter of the largest face of Q fixed by F and wW_{F} corresponds to the w-translate of this face. One description [CD95a] of the Deligne spherical building DSB(A) associated to A is that it is the poset of subsets aW_{F} ⊂ A with a ∈ A and F a proper subset of S, where W ⊃ W_{F} is identified with the set of all atoms in A. Strictly speaking, DSB(A) is not a building, but behaves similarly. The Coxeter sphere embeds naturally into DSB(A) by sending the vertex wW_{F} to the vertex a(w)W_{F} where a(w) is the atom associated with w. This sphere and its translates are the apartments in the Deligne spherical building. The Deligne complex DC(A) is obtained from DSB(A) by coning off each apartment. The following can be shown by examining definitions.

• The cover of DC(A) by the coned off apartments consists of contractible sets with contractible (or empty) intersections.

• The nerve of this cover is the complex X(A).

3 Geometric properties of X(G)

3.1 Geodesics

There is a canonical “combing” of X: for each vertex v ∈ X we have a canonical edge-path from the basepoint ∗ to v. If v = B_{1} · B_{2} · · · · · B_{k} is the normal form without Δ’s, then the edge-path associated to v is

∗, B_{1}, B_{1} · B_{2}, ⋯, B_{1}B_{2} ⋯ B_{k} = v.

Note that left multiplication by Δ sends v to \overline{B}_{1} · \overline{B}_{2} ⋯ \overline{B}_{k} and it sends the combing path from ∗ to v to the combing path from ∗ = Δ(∗) to Δ(v). It follows that by left translating we obtain a canonical path from any vertex to any other vertex. We call all such edge-paths geodesics.

It is interesting that the geodesics are symmetric. This fact follows from the work of R. Charney (Lemma 2.3 of [Cha95]). We will not need it in the sequel, but for completeness we indicate a proof.

Proposition 3.1. The geodesic from v to w is the inverse of the geodesic from w to v.

Proof. First note that an edge-path is a geodesic if and only if every subpath of length 2 is a geodesic. Thus it suffices to prove the Proposition in the case that the geodesic from v to w has length 2. Further, after left-translating, we may assume that the middle vertex of the path is ∗, and so v and w are atoms, say A and B respectively. Denote by C the atom with CA = Δ. Note that C is represented by a geodesic gallery from −A to Q. We now have (recall our convention that we identify an atom A with the chamber π(A)(Q)):

A, ∗, B is a geodesic ⟺ CA = Δ = ∗, C, CB is a geodesic ⟺ C · B is a normal form ⟺ (by Proposition 1.3, see the Figure) no wall W = F ix(s) that abuts Q has Q and −A on one side and B on the other ⟺ (applying “-”) no wall that abuts −Q has −Q and A on one side and −B on the other ⟺ (using that a wall abuts Q iff it abuts −Q and it always separates the two) no wall that abuts Q has Q and −B on one side and A on the other.

If W exists, C · B is not a normal form – C can be extended to a geodesic gallery C′ = Cs with C′ < CB

But the last statement is exactly the same as a previous statement with the roles of A and B interchanged.

3.2 Worddistance

By the wordnorm of a vertex v ∈ V, denoted ||v||, we mean the word-length of the special representative of v, or equivalently, the word-length of the normal form of any representative of v after discarding the Δ’s. Observe that left translation by Δ is wordnorm preserving (it sends B_{1}·B_{2}·· · ··B_{m} to \overline{B}_{1} · \overline{B}_{2} · · · · · \overline{B}_{m}), and so there is a unique function d_{wd} : V × V → ℤ_{+} that is left-invariant and d_{wd}(∗, v) = ||v|| for all v ∈ V. We call d_{wd} the worddistance, but we caution the reader that d_{wd} is not symmetric. For example, if B is an atom, then d_{wd}(∗, B) is the word-length of B, while d_{wd}(B, ∗) is the word-length of the atom C complementary to B (i.e. such that CB = Δ). This observation generalizes as follows.

Lemma 3.2. d_{wd}(v, w) + d_{wd}(w, v) = d_{at}(v, w)δ, where δ is the word-length of Δ.

Proof. After left-translating, we may assume that v = ∗. If w = B_{1} · B_{2} · · · · · B_{k} (without Δ’s), then d_{at}(∗, w) = k and d_{wd}(∗, w) is the word-length of B_{1} ⋯ B_{k}. To compute d_{wd}(w, ∗), denote by C_{i} the atom with C_{i}B_{i} = Δ. Left translate by ⋯ C_{3}\overline{C}_{2}C_{1} to get d_{wd}(w, ∗) = d_{wd}(Δ^{k}, ⋯ C_{3}\overline{C}_{2}C_{1}) = || ⋯ C_{3}\overline{C}_{2}C_{1}||. Since the word-length of C_{i} is obtained from δ by subtracting the word-length of B_{i}, the word-length of ⋯ C_{3}\overline{C}_{2}C_{1} equals kδ − d_{wd}(∗, w), so it remains to argue that the normal form of ⋯ C_{3}\overline{C}_{2}C_{1} does not have any Δ’s. If it did, then the equation

(⋯ C_{3}\overline{C}_{2}C_{1})(B_{1} ⋯ B_{k}) = Δ^{k}

would imply (after cancelling Δ’s) that W B_{1} ⋯ B_{k} = Δ^{l} for some W ∈ A^{+} with atomnorm l < k. We can then find W′ ∈ A^{+} also with atomnorm ≤ l such that W W′ = Δ^{l} (W′ is the concatenation of atoms, possibly with a “bar”, that make the normal form of W). But then the cancellation law implies that B_{1} ⋯ B_{k} = W′Δ^{k−l} has atomnorm ≤ l < k, a contradiction.

The triangle inequality d_{wd}(v, w)+d_{wd}(w, z) ≥ d_{wd}(v, z) is immediate, and so are the inequalities d_{at}(v, w) ≤ d_{wd}(v, w) ≤ δd_{at}(v, w), d_{wd}(w, v) ≤ δd_{wd}(v, w) (showing that d_{wd} is “quasi-symmetric”, and that d_{wd} and d_{at} are quasi-isometric).

We will next argue that the wordnorms of adjacent vertices are always distinct. To this end, we examine the relationship between the normal forms associated to two adjacent vertices, say v = B_{1} · B_{2} · · · · · B_{k} and w = C_{1} · C_{2} · · · · · C_{l} (without Δ’s). Since d_{at}(v, w) = 1, there is an atom X ≠ 1, Δ such that vX and w represent the same coset, i.e.

vX = wΔ^{m}

for some integer m. Similarly, there is a atom Y ≠ 1, Δ such that

wY = vΔ^{n}

for an integer n.

Lemma 3.3. One of m, n is 0 and the other is 1.

Proof. Substituting v = wY Δ^{−n} into vX = wΔ^{m} yields Y Δ^{−n}X = Δ^{m}. If n is even, this equation implies that Y X is a power of Δ, hence Y X = Δ and m+n = 1. If n is odd, then similarly \overline{Y} X = Δ and again m + n = 1.

It remains to argue that m, n ≥ 0. Assume n < 0. Then v = wY Δ^{−n} will have Δ’s in its normal form (as Δ can be moved to the front), contradiction. Similarly m ≥ 0.

It now follows that either ||v|| < ||w|| (if m = 0) or ||w|| < ||v|| (if n = 0). We orient the edge [v, w] from v to w if ||v|| < ||w||. Thus the arrows point away from the basepoint. Note that these orientations are not group invariant.

It is clear from the definition that there are no oriented cycles. This implies that the dimension of X(G) is length(Δ) − 1. Indeed, the set of vertices of a simplex of X(G) is totally ordered and thus a simplex that, say, contains the base point ∗ corresponds to a chain of atoms ∗ < B_{1} < B_{2} < ⋯ < B_{m} of maximal possible length length(Δ).

The following lemma sharpens Lemma 2.1.

Lemma 3.4. Suppose g = X_{1}X_{2} ⋯ X_{k} ∈ A^{+} is written as the product of k atoms, and let g = B_{1} · B_{2} · · · · · B_{l} be the normal form for g. Then l ≤ k and X_{1}X_{2} ⋯ X_{i} < B_{1}B_{2} ⋯ B_{i} for all 1 ≤ i ≤ l.

Proof. That l ≤ k is the content of Lemma 2.1. That X_{1} < B_{1} follows from the definition of the normal form. Assume inductively that X_{1} ⋯ X_{i−1} < B_{1} ⋯ B_{i−1}. Thus we can write B_{1} ⋯ B_{i−1} = X_{1} ⋯ X_{i−1}y for some y ∈ A^{+}, and thus X_{i}X_{i+1} ⋯ X_{k} = yB_{i}B_{i+1} ⋯ B_{l}. But then

X_{i} < α(yB_{i}B_{i+1} ⋯ B_{l}) = α(yB_{i}) < yB_{i}

so that multiplication on the left by X_{1} ⋯ X_{i−1} yields X_{1} ⋯ X_{i} < B_{1} ⋯ B_{i}. The equality in the displayed formula follows from Proposition 1.2.

3.3 Contractibility of X(G) and its topology at infinity

Denote by d = card(S) − 1 the dimension of the Coxeter sphere (the unit sphere in V_{W}). In this section we prove:

Theorem 3.5. X(G) is contractible and it is proper homotopy equivalent to a cell complex which is d-dimensional and (d − 2)-connected at infinity.

In particular, this recovers a theorem of C.C. Squier [Squ94]: G is a virtual duality group of dimension d and A is a duality group of dimension d + 1.

The method of proof is to consider the function F : X → [0, ∞) that to a vertex v assigns the wordnorm F (v) = ||v|| and is linear on each simplex. The theorem follows by standard methods (see [BB97], [BF97]) from:

Proposition 3.6. F is nonconstant on each edge of X(G). The ascending link at each vertex v is either contractible or homotopy equivalent to the sphere S^{d−1} and the descending link at each vertex v ≠ ∗ is contractible.

For a vertex v, the ascending link Lk_{↑}(v) [descending link Lk_{↓}(v)] is defined to be the link of v in the subcomplex of X(G) spanned by the vertices w with F (w) ≥ F (v) [F (w) ≤ F (v)].

We need some lemmas.

Lemma 3.7. Given atoms A and B there is a unique atom C such that for every u ∈ A^{+}

A < Bu ⟺ C < u.

Proof. First note that by Proposition 1.2 A < Bu is equivalent to A < Bα(u) and C < u is equivalent to C < α(u), so without loss of generality we may assume that u = U is an atom. Consider the set Ξ of atoms U with A < BU. Note that Ξ contains the atom B^{∗} with BB^{∗} = Δ. Denote by Ω the set of atoms D such that D < U for each U ∈ Ξ. Observe the following properties of Ω:

• Ω contains the trivial atom (that maps to 1 ∈ W).

• If D ∈ Ω and D′ < D then D′ ∈ Ω.

• If σ, τ are two generators and if D < Dσ ∈ Ω, D < Dτ ∈ Ω, then Dprod(σ, τ, m(σ, τ)) ∈ Ω.

By Proposition 1.1, there is a unique atom C such that

D ∈ Ω ⟺ D < C.

In particular, C ∈ Ω, so A < BU implies C < U and it remains to prove the reverse implication, i.e. it remains to show that A < BC. Note that BC is an atom, since C ∈ Ω implies that C < B^{∗} so BC < BB^{∗} = Δ. Let Y = A ∨ BC be the smallest atom with A < Y and BC < Y. Thus we can write Y = BCD for an atom D and we have to show that D is the trivial atom. This will be accomplished if we show that CD ∈ Ω, by the choice of C. Thus we need to argue that CD < U whenever A < BU. But A < BU implies C < U and thus BC < BU, so BU is a gallery larger than both A and BC. It follows that the atom α(BU) is also larger than both A and BC, so by the choice of Y we have Y = BCD < α(BU) < BU, and cancelling B we obtain CD < U.

Lemma 3.8. Given an atom A and an element b ∈ A^{+} there is a unique atom C such that for every u ∈ A^{+}

A < bu ⟺ C < u.

Proof. We induct on the number k of atoms in the normal form b = B_{1} · B_{2} · · · · · B_{k} for b. When k = 1 this is Lemma 3.7. Suppose k > 1. Applying induction to B_{1} ⋯ B_{k−1}, we find an atom C′ such that for all w ∈ A^{+}

A < B_{1} ⋯ B_{k−1}w ⟺ C′ < w.

Lemma 3.7 now gives an atom C such that

C′ < B_{k}u ⟺ C < u

for all u ∈ A^{+}. We now have

A < B_{1} ⋯ B_{k}u ⟺ C′ < B_{k}u ⟺ C < u.

Proof of Proposition 3.6. Represent v as a normal form B_{1}·B_{2}·· · ··B_{k} without Δ’s. A representative of an adjacent vertex can be obtained from B_{1} ⋯ B_{k} by multiplying on the right by an atom, say B. Note that F (vB) < F (v) precisely when Δ < B_{1} ⋯ B_{k}B. By Lemma 3.8, there is an atom C such that for any atom B

Δ < B_{1} ⋯ B_{k}B ⟺ C < B.

Thus the descending link at v can be identified with the poset of atoms B ≠ 1, Δ such that C < B and the ascending link with the poset of atoms B ≠ 1, Δ such that C ≮ B. The former is a cone with cone-point C (unless C = Δ which corresponds to v = ∗). To understand the latter, for each atom D consider the subset S(D) = ∪{B|B < D} of the Coxeter sphere, where we now identify an atom B with the intersection of the chamber π(B)(Q) with the unit sphere in V_{W}. The complement of 1 ∪ ∪{B|C < B} is covered by the intersections with the sets S(D) for C ≮ D. Proposition 1.1 implies that the collection of S(D)’s is closed under nonempty intersections. The elements are contractible ^{1}, and thus the poset of the cover (which can be identified with the ascending link) is homotopy equivalent to the underlying space S^{d} \(1∪∪{B|C < B}). If the balls 1 and ∪{B|C < B} are disjoint, then this complement is homotopy equivalent to S^{d−1} and if they intersect then the complement is contractible (the two balls are convex).

Remark 3.9. The proof also shows that the dualizing module H_{c}^{d}(X(G); ℤ) is a free abelian group. In this situation, H_{c}^{d}(X(G); ℤ) is isomorphic to

⨁_{v∈V} H^{d−1}(Lk_{↑}(v, X(G)); ℤ).

For details see [BF97]. Remark 3.10. There is another proof of the contractibility of X(G) that follows the lines of Charney’s Fellow Traveller Property [Cha92]. One shows that the function ϕ : V → V defined by ϕ(∗) = ∗ and

ϕ(A_{1} · A_{2} · · · · · A_{k−1} · A_{k}) = A_{1} · A_{2} · · · · · A_{k−1}

extends to a simplicial map X(G) → X(G) which is homotopic to the identity. Then one observes that the image of any finite subcomplex of X(G) under a high iterate of ϕ is a point. Remark 3.11. The Morse function F : X(G) → [0, ∞) gives rise to another normal form in A^{+} (and a combing of X(G)). For a vertex v = v_{0} ≠ ∗ define v_{1} to be the vertex determined by the cone point in the descending link at v. In other words, v_{1} = vC where C is the atom defined by Δ < vX ⟺ C < X. Now define v_{2} in the same way starting at v_{1} etc. until ∗ is reached. The sequence of the vertices constructed (read in reverse) determine a normal form. It can be shown that this normal form is symmetric and locally determined (an edge-path is a left translate of a normal form iff every two-edge subpath is). In fact, the path A, ∗, B is a translate of a normal form iff A ∨ B = Δ. (Proof: Let ^{∗}A be the atom with ^{∗}AA = Δ and let B^{∗} be the atom with BB^{∗} = Δ. Then our path is a translate of ∗,^{∗} A,^{∗} AB and the condition is Δ <^{∗} ABX ⟺ B^{∗} < X. We have:

Δ <^{∗} ABX ⟺ A < BX ⟺ A ∨ B < BX.

If A ∨ B = Δ, the latter condition is equivalent to B^{∗} < X and if A ∨ B ≠ Δ, then B^{∗} < X is a strictly weaker condition on X.) This is to be contrasted with the fact [Cha95] that A, ∗, B is a translate of a left-greedy normal form iff A ∧ B = 1. Remark 3.12. The connectivity at infinity of right-angled Artin groups was studied by N. Brady and Meier [BMa]. These groups act cocompactly on CAT(0) spaces and the distance from a point is a good Morse function in that context. Brown and Meier [BMb] analyze which Artin groups of infinite type are duality groups.

1This can be seen by representing S(D) \ 1 as the union of sets S(E, D) \ 1 for 1 ≠ E < D, where S(E, D) = ∪{F |E < F < D} and observing that the intersections with Y = S^{d} \ (1 ∪ ∪{B|C < B}) form an intersection-cover of S(D) ∩ Y whose poset is a cone

3.4 NPC

We wish to study the properties of the distance function on X(G), and in particular its “nonpositively curved” aspects. We start by studying triangles formed by geodesics. It is convenient to translate the triangle so that one vertex is ∗. Denote the other two vertices by v and w.

Proposition 3.13. All edges in the geodesic from v to w whose orientation points towards v occur at the beginning of the path, and these are followed by the edges oriented towards w.

In other words, the wordnorm first decreases and then increases along the geodesic.

A geodesic triangle in X(G)

Proof. Suppose on the contrary that x, y, z are (special representatives of) consecutive vertices on the geodesic from v to w and that ||x|| < ||y|| > ||z||. Thus there are atoms B and C such that xB = y and yC = zΔ. It follows that the first atom in the normal form for x(BC) = zΔ = Δ\overline{z} is Δ. Since B · C is a normal form, Proposition 1.2 implies that the first atom in the normal form for xB = y is Δ, a contradiction.

The following is the key “non-positively curved” feature of the complex X(G). There are two versions, one for each metric. The d_{wd}-version is more useful, since the inequality is strict. It will allow us to run Cartan’s fixed-point argument.

Theorem 3.14. Suppose p is a vertex on the geodesic from v to w and v ≠ p ≠ w. Let u be any vertex. Then

d_{at}(u, p) ≤ max{d_{at}(u, v), d_{at}(u, w)

and

d_{wd}(u, p) < max{d_{wd}(u, v), d_{wd}(u, w)}.

Moreover,

d_{wd}(u, p) ≤ max{d_{wd}(u, v), d_{wd}(u, w)} − min{d_{at}(v, p), d_{at}(w, p)}.

Proof. Without loss of generality u = ∗ and then the proof follows from Proposition 3.13.

We remark that the function d_{wd}(∗, ·) is not necessarily convex along [v, w], cf. A = ℤ × ℤ × ℤ.

3.5 Centers

Let T be a finite set of vertices of X(G). The circumscribed radius r(T) is the smallest integer r such that for some vertex v ∈ X, r = max_{t∈T} d_{wd}(t, v). (Note the order of points t and v here.) A vertex v is said to be a center of T if r(T) = max_{t∈T} d_{wd}(t, v). A set might have more than one center, but we have the next best thing.

Proposition 3.15. If v and w are two centers of T, then v and w span an edge.

Proof. Suppose the geodesic from v to w passes through another vertex z. If t ∈ T, then

d_{wd}(t, z) < max{d_{wd}(t, v), d_{wd}(t, w)} ≤ r(T)

contradicting the definition of r(T).

We next have the Cartan fixed point theorem in our setting.

Theorem 3.16. Every finite subgroup H < G leaves a simplex of X(G) invariant (and fixes its barycenter). In particular, there are only finitely many conjugacy classes of finite subgroups in G.

Proof. Let T be an orbit of H. The set of centers of T is H-invariant and spans a simplex in X(G).

3.6 Minsets of elements

In the theory of isometric actions on nonpositively curved spaces an important role is played by the minsets of isometries. These are the sets of points that move the least under the given isometry. Here we explore the analogous concept using the worddistance.

Proposition 3.17. Let g ∈ G and suppose v, w are two vertices of X(G) and d_{wd}(v, g(v)) ≥ d_{wd}(w, g(w)). Then for each vertex z on the geodesic from v to w we have d_{wd}(z, g(z)) ≤ d_{wd}(v, g(v)).

Proof. We may assume that v = ∗, by left-translating (and replacing g by a conjugate). Let w = X_{1}·X_{2}·· · ··X_{k} be the normal form without Δ’s. The element g has the form g = B_{1}·B_{2}·· · ··B_{m} (no Δ’s) or g = B_{1} · B_{2} · · · · · B_{m}Δ.

Case 1. g = B_{1} · B_{2} · · · · · B_{m}.

Note that the vertex g(w) can be written as B_{1} ⋯ B_{m}X_{1} ⋯ X_{k}, and also as X_{1} ⋯ X_{k}u where ||u|| = d_{wd}(w, g(w)). Since we are assuming that ||u|| ≤ ||g|| = d_{wd}(∗, g(∗)), it follows that we have equality in A^{+}:

B_{1} ⋯ B_{m}X_{1} ⋯ X_{k} = X_{1} ⋯ X_{k}uΔ^{r}

for some r ≥ 0. In particular, X_{1} ⋯ X_{i} < B_{1} ⋯ B_{m}X_{1} ⋯ X_{i} (by Proposition 1.4) and so we have

B_{1} ⋯ B_{m}X_{1} ⋯ X_{i} = X_{1} ⋯ X_{i}u_{i}

for some u_{i} ∈ A^{+}. In particular,

d_{wd}(X_{1} ⋯ X_{i}, g(X_{1} ⋯ X_{i})) ≤ ||u_{i}|| = ||B_{1} ⋯ B_{m}|| = ||g|| = d_{wd}(∗, g(∗)).

Case 2. g = B_{1} · B_{2} · · · · · B_{m}Δ.

The calculation is similar. We now get

B_{1} ⋯ B_{m}ΔX_{1} ⋯ X_{k} = X_{1} ⋯ X_{k}uΔ^{r}

and r ≥ 1 since ||u|| ≤ ||g|| = ||B_{1} ⋯ B_{m}||. Now push the lone Δ on the left-hand side all the way to the right and cancel it to get B_{1} ⋯ B_{m}\overline{X}_{1} ⋯ \overline{X}_{k} = X_{1} ⋯ X_{k}uΔ^{r−1}. Again by Proposition 1.4 we get

X_{1} ⋯ X_{i} < B_{1} ⋯ B_{m}\overline{X}_{1} ⋯ \overline{X}_{i}

so that

B_{1} ⋯ B_{m}\overline{X}_{1} ⋯ \overline{X}_{i} = X_{1} ⋯ X_{i}u_{i}

for some u_{i} ∈ A^{+} with ||u_{i}|| = ||B_{1} ⋯ B_{m}|| = ||g||. Passing again to A/ < Δ > we obtain g(X_{1} ⋯ X_{i}) = X_{1} ⋯ X_{i}u_{i} and so

d_{wd}(X_{1} ⋯ X_{i}, g(X_{1} ⋯ X_{i})) ≤ ||u_{i}|| = ||g||.

If g ∈ G and d ∈ ℤ, let T_{d}(g) = {v ∈ X^{(0)}|d_{wd}(v, g(v)) ≤ d}. Recall that d_{wd} is not symmetric and note that writing d_{wd}(g(v), v) instead in the definition would give T_{d}(g^{−1}). The minimal d with T_{d}(g) ≠ ∅ is the displacement length of g, denoted λ(g). The set T_{d}(g) for this d is the minset of g, denoted A_{g}.

A set T of vertices is starlike with respect to t_{0} ∈ T if for every t ∈ T all vertices on the geodesic from t_{0} to t are in T. It can be shown, by the argument in Remark 3.10, that the span of a starlike set is contractible. A set of vertices is convex if it is starlike with respect to each of its vertices.

Corollary 3.18. The minset A_{g} of each g ∈ G is convex.

Example 3.19. Suppose H < G is a finite subgroup. The set of vertices of H-invariant simplices can be described as

⋂_{h∈H} A_{h}

and this set is therefore convex. This implies that F ix(H) ⊂ X(G) is contractible: Consider the cover of F ix(H) by H-invariant simplices. Since F ix(H) intersects each such simplex in a contractible set, the nerve of the collection of H-invariant simplices and the nerve of this cover are the same. So F ix(H) has the homotopy type of the union of H-invariant simplices, which is contractible by the above discussion.

A similar argument shows the following:

Lemma 3.20. Let g ∈ A^{+} have normal form B_{1} · B_{2} · · · · · B_{k}. Also suppose that the displacement of ∗ is not smaller than the displacement of the vertex v with special representative D_{1} · D_{2} · · · · · D_{l}, i.e. that d_{wd}(∗, g(∗)) ≥ d_{wd}(v, g(v)). Then D_{1}D_{2} ⋯ D_{l} < B_{1}B_{2} ⋯ B_{k}D_{1}D_{2} ⋯ D_{l}.

Proof. Let D̃ = D_{1}D_{2} ⋯ D_{l}. d_{wd}(D_{1} ⋯ D_{k}, gD_{1} ⋯ D_{l}) is computed as follows: let Ẽ = E_{1} · E_{2} · ⋯ E_{l} be the normal form with ẼD̃ = Δ^{l}, find the normal form of Ẽ(B_{1} ⋯ B_{k}D̃), discard Δ’s and count letters. Since d_{wd}(∗, g(∗)) ≥ d_{wd}(v, g(v)), at least l Δ’s will have to be discarded, i.e. Δ^{l} < ẼB_{1}B_{2} ⋯ B_{k}D̃. Write Δ^{l} = ẼD̃ and cancel Ẽ to obtain D̃ < B_{1}B_{2} ⋯ B_{k}D̃.

4 Applications

We now turn to the group-theoretic consequences of the discussion of the geometric properties of X(G).

4.1 Translation lengths

The translation length of g ∈ G, denoted τ(g), is defined to be

τ(g) = lim_{n→∞} \frac{||g^{n}(∗)||}{n}.

See [GS91] for basic properties of translation functions (in the more general setting of finitely generated groups). An important feature of τ is that conjugate elements have the same translation length and the restriction of τ to a torsion-free abelian subgroup is a (non-symmetric) semi-norm. In [GS91], Gersten and Short prove that in biautomatic groups elements of infinite order have nonzero translation lengths, and this gives rise to many properties of such groups (e.g. a biautomatic group cannot contain the Heisenberg group as a subgroup). Finite type Artin groups are biautomatic by [Cha92]. Here we show that translation lengths of torsion-free elements in G are bounded away from 0, and this gives rise to additional group-theoretic properties.

Let g ∈ G be an element which is not conjugate to an atom, or an atom followed by Δ. By T_{n} denote the set {∗, g(∗), g^{2}(∗), ⋯, g^{n−1}(∗)}. Let r_{n} be the circumscribed radius of S_{n}. Since T_{1} ⊂ T_{2} ⊂ ⋯ we have r_{1} ≤ r_{2} ≤ · · ·. Suppose that r_{n} = r_{n+1} for some n. Let z be a center for T_{n+1}. Then the ball of radius r_{n} = r_{n+1} centered at z covers both T_{n} and g(T_{n}) and so z is a center for both. But then g(z) is also a center for g(T_{n}) and we conclude that z and g(z) span an edge, so that g is conjugate to an atom or an atom followed by Δ. Thus r_{n} ≥ n for all n.

We now see that max_{i≤n} d_{wd}(∗, g^{i}(∗)) ≥ n, and in particular there are infinitely many n such that d_{wd}(∗, g^{n}(∗)) ≥ n. It follows that the translation length τ(g) ≥ 1.

Recall that δ = length(Δ).

Theorem 4.1. The set of translation lengths of elements of G of infinite order is bounded below by \frac{1}{2δ}.

Proof. We have seen that if g is of infinite order and not conjugate to B or BΔ for an atom B, then τ(g) ≥ 1. Thus if g^{2δ} is not conjugate to an atom or its inverse, then τ(g) = \frac{1}{2δ} τ(g^{2δ}) ≥ \frac{1}{2δ}.

The length homomorphism length : A → ℤ descends to the homomorphism length : G → ℤ/2δ. Since conjugate elements have the same length, length(g^{2δ}) = 0 ∈ ℤ/2δ, and the only atom with 0 length is the identity element, it follows that if g^{2δ} is conjugate to an atom or its inverse, then g has finite order.

Corollary 4.2. Every abelian subgroup A of G (or A) is finitely generated.

Proof. Since G is virtually torsion-free, after passing to a finite index subgroup of A we may assume that A is torsion-free. Since the virtual cohomological dimension of G is finite, it follows that A is a subgroup of the finite-dimensional vector space W = A ⊗ ℚ. Since τ : A → [0, ∞) extends to a (non-symmetric) norm on W and its values on A \ {0} are bounded away from 0, it follows that A is discrete in W. (Strictly speaking, this is not a norm since τ is not symmetric. We could symmetrize, or else work with a non-symmetric norm which is just as good in the present context.)

Corollary 4.3. G does not have any infinitely divisible elements of infinite order.

The following corollary is a consequence of Theorem 3.5 of G. Conner’s work on translation lengths [Con]. For completeness, we sketch the proof.

Corollary 4.4. Every solvable subgroup of G (or A) is virtually abelian.

Proof. Let H be a solvable subgroup of G. Since G is virtually torsion-free, we may assume that H is torsion-free after replacing it with a subgroup of finite index. By induction on the length of the derived series, the commutator subgroup [H, H] is virtually abelian, and hence finitely generated. Let K ⊂ [H, H] be an abelian subgroup of finite index which is characteristic in [H, H]. The conjugation action of H on K is virtually trivial, since it preserves translation lengths (the image of the homomorphism H → GL(K) consists of elements of finite order and is therefore finite). Thus the action of H on [H, H] is virtually trivial as well. After passing to a subgroup of H of finite index, we may assume that this action is trivial. The proof concludes by an argument in [GS91]. We now claim that [H, H] is the trivial group, and so (the new) H is abelian. Indeed, suppose that x, y ∈ H and z = [x, y] = xyx^{−1}y^{−1} ≠ 1. By our construction, z commutes with both x and y, and it has infinite order. We compute that [x^{n}, y^{n}] = z^{n^{2}} and so the translation length of z is 0, a contradiction.

4.2 Finite subgroups of G

It is well-known to the experts that all finite subgroups of G are cyclic, and in fact the kernel of the homomorphism G → ℤ/2δ is torsion-free, where δ = ||Δ|| and the homomorphism is the length modulo 2δ. (Note that the argument of Theorem 4.1 gives another proof of this fact.)

In this section we will use the geometric structure of X(G) to give a classification of finite subgroups of G up to conjugacy.

Theorem 4.5. Every finite subgroup H < G is cyclic. Moreover, after conjugation, H transitively permutes the vertices of a simplex σ ⊂ X(G) that contains ∗ and H has one of the following two forms:

Type 1: The order of H is even, say 2m. It is generated by an atom B. The vertices of σ are ∗, B, B^{2}, ⋯, B^{m−1} (all atoms) and B^{m} = Δ. Necessarily, \overline{B} = B (since Δ fixes the whole simplex).

Type 2: The order m of H is odd, the group is generated by BΔ for an atom B, and the vertices ∗, B, B\overline{B}, B\overline{B}B, ⋯, (B\overline{B})^{(m−1)/2}B (all atoms) are permuted cyclically and faithfully by the group (so the dimension of σ is m − 1). Since m is odd, the square B\overline{B} of the generator also generates H.

An example of a type 1 group is < B > for B = σ_{1}σ_{3}σ_{2} in the braid group B_{4}/ < Δ^{2} > (of order 4). An example of a type 2 group is < σ_{1}Δ > in B_{3}/ < Δ^{2} > (of order 3).

The key to this is:

Lemma 4.6. The set of vertices of any simplex σ in X admits a cyclic order that is preserved by the stabilizer Stab(σ) < G.

Proof. We can translate σ so that ∗ is one of its vertices. Let the cyclic order be induced from the linear order ∗ < B_{1} < B_{2} < ⋯ < B_{k} given by the orientations of the edges of σ (equivalently, by the lengths of the atoms B_{i}). We need to argue that the left translation by B_{i}^{−1} is going to produce the same cyclic order. We can write B_{i+1} = B_{i}C_{i} for atoms C_{i}, so that the vertices of σ are ∗, C_{1}, C_{1}C_{2}, ⋯, C_{1} ⋯ C_{k}. After translation by B_{1}^{−1} the vertices are B_{1}^{−1}, ∗, C_{2}, C_{2}C_{3}, ⋯, C_{2} ⋯ C_{k}, where B_{1}^{−1} should be replaced by the special representative of the coset B_{1}^{−1} < Δ >. Let Y be the atom with C_{1}C_{2} ⋯ C_{k}Y = Δ. Then B_{1}^{−1}Δ = C_{1}^{−1}Δ = C_{2}C_{3} ⋯ C_{k}Y and C_{2}C_{3} ⋯ C_{k}Y is this canonical representative (it is a subword of Δ, so it is an atom). Since C_{2} ⋯ C_{k} < C_{2} ⋯ C_{k}Y, it follows that the induced ordering on the vertices is a cyclic permutation of the old one. Repeating this i times gives the claim.

Proof of Theorem 4.5. Recall that by Corollary 3.16 there is an H-invariant simplex σ ⊂ X(G). By passing to a face if necessary and conjugating, we may assume that H acts transitively on the vertices of σ, and that ∗ is a vertex of σ. Say there are m vertices and choose h ∈ H that rotates the simplex by one unit. Note that h^{m} is either 1 or Δ. Also, h is either an atom B or else h = BΔ for an atom B. In the first case, the vertices of the simplex are ∗, B, BB, ⋯, B^{m−1} and these are all atoms (since the arrow points from B^{i} to B^{i+1} for each i = 0, 1, ⋯, m − 2). Thus 1 < B^{m−1} < Δ and hence B < B^{m} < BΔ, so it follows that B^{m} = Δ, and H is of type 1. In the second case, the vertices of σ are ∗, B, B\overline{B}, B\overline{B}B, ⋯ and they are similarly all atoms. If m is even, the last vertex in this sequence is (B\overline{B})^{m/2−1}B, and this is still an atom. It follows that (B\overline{B})^{m/2} = Δ. But then h^{m} = Δ ∈ H fixes the whole simplex and in particular B = \overline{B} and H is of type 1. So suppose m is odd. Then the last vertex of σ is (B\overline{B})^{\frac{m−1}{2}} and we have (B\overline{B})^{\frac{m−1}{2}} B = Δ. Multiplication by Δ on the right reveals that h^{m} = 1 and H is of type 2.

4.3 Normal abelian subgroups and centers of finite index subgroups

We use minsets to prove:

Theorem 4.7. Assume that the associated Coxeter group W is nonabelian and irreducibe. The action of G on itself by conjugation does not have nontrivial finite orbits. (Singletons consisting of central elements are trivial orbits.) In particular, the center of any finite index subgroup of G is either trivial or < Δ > (if the latter is central in G).

We remark that Δ is central in G if and only if it is central in A (if Δg = gΔ in G then a priori we only get Δg = gΔ^{2m+1} in A, but m must be 0 by length considerations) if and only if the bar involution is trivial.

We now turn to the lemmas needed in the proof.

Lemma 4.8. Suppose A_{1}, ⋯, A_{p} is a finite collection of convex sets in V = G/ < Δ > permuted by left translations. Then each A_{i} is empty or all of V.

Proof. If not, we may assume ∅ ≠ A_{i} ≠ V for each i, by discarding the copies of ∅ and V from the collection. Let q ≥ 1 be the largest integer such that for some i_{1} < i_{2} < ⋯ < i_{q} the intersection

⋂_{j=1}^{q} A_{i_{j}} ≠ ∅.

Now pass to the collection of q-fold intersections of the A_{i}’s. We can therefore assume that the sets in the collection, still denoted A_{i}, are pairwise disjoint. Since G acts transitively on V, the sets A_{i} must cover V. Say ∗ ∈ A_{1}. Let σ be a generator of A^{+} and consider the line L = {σ^{j}(∗)|j ∈ ℤ}. By convexity, if σ^{j}(∗), σ^{l}(∗) ∈ A_{i}, then σ^{k}(∗) ∈ A_{i} for k between j and l (this is because the normal form of σσ ⋯ σ is σ · σ · · · · · σ). We conclude that for j large each σ^{j}(∗) belongs to the same A_{i}. Left translation by σ^{−j} takes σ^{j}(∗) ∈ A_{i} to ∗ ∈ A_{1} and it takes σ^{2j}(∗) ∈ A_{i} to σ^{j}(∗) ∈ A_{i}, and so i = 1. Convexity now implies that σ^{j}(∗) ∈ A_{1} for all j ≥ 0, and in particular σ(∗) ∈ A_{1}.

Let v be now any vertex and assume v ∈ A_{1}. Choose g ∈ G with g(v) = ∗. Thus g(A_{1}) = A_{1} and we then have g^{−1}(σ(∗)) ∈ A_{1}. The two choices of g give vσ ∈ A_{1} and v\overline{σ} ∈ A_{1}. Since any vertex can be reached from ∗ by successively right-multiplying by a generator, we conclude A_{1} = V, a contradiction.

It is convenient to introduce the following notion. Recall that x_{1} ·x_{2} ⋯ x_{k} is a normal form if and only if x_{i} = α(x_{i}x_{i+1}) for all i. Motivated by this observation we construct, following R. Charney [Cha92], a finite graph whose vertex set is the set of atoms not equal to 1 or Δ, and there is an arrow from x to y if x = α(xy). A nontrivial normal form without Δ’s is simply a finite directed path in this graph. We refer to this graph as the Charney graph. So the elements of V are in 1-1 correspondence with oriented paths in the Charney graph (with ∗ corresponding to the empty path, and atoms corresponding to one point paths).

Proposition 4.9. If W is irreducible, then any two atoms x, y not equal to 1 or Δ can be joined by an oriented path in the Charney graph.

Proof. As the first case, we assume that x and y are generators. Let x = g_{1}, g_{2}, ⋯ g_{k} = y be a sequence of generators such that successive elements do not commute (this sequence exists by the irreducibility assumption). Then

g_{1} · g_{1}g_{2} · g_{2} · g_{2}g_{3} · g_{3} ⋯ g_{k−1} · g_{k−1}g_{k} · g_{k}

is the desired path.

Next, we observe that if x ≠ 1 is any atom, then there is a generator g such that x · g is a normal form. Simply take g to be the last generator in a word representing x.

It remains to argue that if y is any atom not equal to 1 or Δ, there is a generator g and an oriented path from g to y. Let S be the set of generators a such that some word representing y begins with a. If S consists of a single element a, then a · g is a normal form and we are done. We proceed by induction on the cardinality of S. We first note that S is a proper subset of the generating set (if a chamber is separated from the fundamental chamber Q by every wall adjacent to Q, then the chamber is the antipodal chamber −Q). Next, observe that Δ(S) · y is a normal form, where Δ(S) is the Δ-element in the subgroup generated by S. Let c be a generator not in S. Then cΔ(S) is an atom that can begin with c or with an element of S that commutes with c (this follows from the cancellation law and the fact that if y begins with c and x then it begins with prod(c, x, m(c, x))). Since cΔ(S) · Δ(S) · y is a normal form, we have replaced the original set S by the set S′ consisting of c and the elements of S that commute with c.

Case 1. c can be chosen so that at least two elements of S don’t commute with c.

Then card(S′) < card(S) and we are done by induction.

Recall that the Coxeter graph is a tree, and consider the forest spanned by S. Two generators commute if and only if they are not adjacent in the Coxeter graph.

Case 2. The forest has more than one component.

If there are two components separated by a single vertex c, then clearly card(S′) < card(S) and we are done. Otherwise choose c to be adjacent to one component while separating it from another, and so that the distance between the two components is as small as possible. Then S′ has the same cardinality as S, but the associated forest has two components that are closer together than in the old forest. Repeating this procedure eventually produces two components separated by a single vertex.

Case 3. The forest is a tree and card(S) > 1.

Then choose c to be adjacent to a vertex in the tree. S′ has the same cardinality as S but the underlying forest has > 1 component.

Lemma 4.10. Suppose W is irreducible. Then for any chamber R and any wall W that abuts R there is a normal form D_{1} · D_{2} · · · · · D_{l} such that

• D_{l} is a single generator,

• The gallery associated to D_{1}D_{2} ⋯ D_{l} that starts at the fundamental chamber Q ends at the chamber R and last crosses wall W.

Moreover, if B ≠ 1, Δ is a given atom, the normal form can be chosen so that B · D_{1} is a normal form.

Proof. Using the connectivity of the Charney graph, start with a normal form E_{1} · E_{2} · · · · · E_{m} so that E_{m} is a single generator (and B · E_{1} is a normal form). Say R′ is the terminal chamber of the gallery E_{1} ⋯ E_{l} and the last wall crossed is W′. If X is a generator that does not commute with D_{l}, then D_{1} · D_{2} · · · · · D_{l} · D_{l} · (D_{l}X) · X is also a normal form whose gallery ends at R′, but the last wall crossed is a different wall from W′. By irreducibility, any two generators can be connected by a sequence of generators with successive generators noncommuting, and thus we can construct a gallery as above that ends at R′ and last crosses a preassigned wall abutting R′. By repeating the last atom in such a normal form, we can construct a similar gallery that ends in any preassigned chamber adjacent to R′ and by iterating these operations we can get to R and W.

Lemma 4.11. If W is irreducible, then A_{g} ≠ V for every nontrivial g ∈ G, unless g = Δ and Δ is central.

We will actually prove a stronger form of this lemma. Consider the set H ⊂ G of elements g ∈ G such that the function v ↦ d_{wd}(v, g(v)) defined on V is bounded above. It is then clear that H is a normal subgroup of G.

Proposition 4.12. If W is irreducible, then H is either trivial or equal to < Δ > (if Δ is central).

Proof. We will first argue that every element of H has finite order. This will imply that H is finite (since H → W will then be injective).

Let g ∈ H have infinite order. Notice that ||g^{i}(∗)|| → ∞ as i → ∞. This is because g^{i}(∗) = g^{j}(∗) for i ≠ j would imply that g^{i−j} equals 1 or Δ, so g would have finite order. Fix a large number N (to be specified later) and replace g by a power if necessary so that ||g(∗)|| > N. Further, by conjugating g if necessary, we may assume that ||g(∗)|| = d_{wd}(∗, g(∗)) realizes the maximum of the (bounded) function v ↦ d_{wd}(v, g(v)).

As usual, we have either g = B_{1} · B_{2} · · · · · B_{k} or g = B_{1} · B_{2} · · · · · B_{k}Δ. Note that k > N/δ is large.

Case 1. g = B_{1} · B_{2} · · · · · B_{k}.

If D̃ = D_{1} · D_{2} · · · · · D_{l} is a normal form such that B_{k} · D_{1} is a normal form, then consider the vertex D_{1} ⋯ D_{p}(∗) = D_{1} ⋯ D_{l}. From Lemma 3.20 we get D_{1}D_{2} ⋯ D_{l} < B_{1}B_{2} ⋯ B_{k}D_{1}D_{2} ⋯ D_{l}. Assume in addition that l ≤ k. Since the right-hand side is also a normal form, Lemma 2.1 implies that D_{1}D_{2} ⋯ D_{l} < B_{1}B_{2} ⋯ B_{l}. We will use this only when l = k. Summarizing, if B_{k} · D_{1}, then D_{1} ⋯ D_{k} < B_{1} ⋯ B_{k}. But we now argue that we can choose D̃ = D_{1} · D_{2} · · · · · D_{k} so that D_{1} follows B_{k} (in the Charney graph) but D̃ ≮ B_{1} ⋯ B_{k}.

Say M is an integer such that the atomnorm of the galleries constructed in Lemma 4.10 is bounded by M.

Each atom B ≠ Δ crosses at most δ − 1 walls in the Coxeter sphere. Thus B_{1} ⋯ B_{k} crosses at most k(δ − 1) walls. Therefore some wall W is crossed by B_{1} ⋯ B_{k} at most k(δ − 1)/δ times and the same is true for any initial piece of B_{1} ⋯ B_{k}. We will get a contradiction (for large N and k) by arguing that D̃ can be chosen so that it crosses a preassigned wall at least k − M times (contradiction arising if k − M > k(δ − 1)/δ i.e. if k > δM, and this can be arranged if N > δ^{2}M).

Start with a normal form D_{1} · D_{2} · · · · · D_{l} with l ≤ M so that B_{k} · D_{1} is a normal form, D_{l} is a single generator and the last wall crossed is W. This is possible by Lemma 4.10 (a left translation may be necessary before applying the lemma since B_{1} ⋯ B_{k} may end at a chamber different from Q). Then set D_{i} = D_{l} for i = l + 1, l + 2, ⋯, k.

Case 2. g = B_{1} ⋯ B_{k}Δ.

This is entirely analogous, except for some overlines, and is left to the reader.

The proof of the proposition then follows from the following result.

Proposition 4.13. If W is irreducible and H < G is a finite normal subgroup, then H is trivial or H =< Δ > with Δ central.

Proof. Since H is finite, there is an H-invariant simplex in X(G). Since H is normal, every vertex belongs to an H-invariant simplex. If H =< Δ >, then ∗ is fixed by H, and hence every vertex is fixed by H. Thus the bar involution is trivial and Δ is central. If H is nontrivial and not equal to < Δ >, then H must contain a nontrivial atom B ≠ Δ (the inverse of an element of the form BΔ is an atom). Left translation by B moves every vertex to an adjacent vertex, and so d_{wd}(v, Bv) < δ for every vertex v. On the other hand, for any g ∈ G the numbers d_{wd}(v, g(v)) are all congruent mod δ to each other, as v ranges over the vertices of X(G), so in our situation we see that all displacements d_{wd}(v, Bv) are equal to each other. By the irreducibility of W, there is a normal form B · B_{1} · B_{2} · · · · · B_{k} with B_{k} any prechosen atom. Lemmas 3.20 and 2.1 now imply that B_{1} ⋯ B_{k} < BB_{1} ⋯ B_{k−1}. In particular, we see that ||B_{k}|| ≤ ||B||. By choosing B_{k} to have length δ − 1, we conclude that B has length δ − 1. There is then a generator σ such that σB = Δ and we conclude that B^{−1} = Δσ ∈ H. But Δσ has infinite order, unless σ and \overline{σ} are the only generators. (To see this, note that X · X · X · · · · · X is a normal form where X = prod(σ, \overline{σ}, m(σ, \overline{σ})).) If σ = \overline{σ}, then there is only one generator and G =< Δ >≅ ℤ_{2} so there is nothing to prove. If σ ≠ \overline{σ}, then each half of the Artin relation σ\overline{σ}σ ⋯ = \overline{σ}σ\overline{σ} ⋯ has length equal to an odd integer m ≥ 3. In particular, σ\overline{σ} ∈ H is a atom, so it must have length δ − 1, and this forces m = 3. We are now reduced to the classical braid group on 3 strands modulo the center. To finish the argument, note that in this case σ\overline{σ} · \overline{σ} is a normal form, while \overline{σ} ≮ σ\overline{σ}.

Proof of Theorem 4.7 Let g_{1}, ⋯, g_{n} be a finite orbit under conjugation. Note that there is K > 0 such that d_{wd}(v, g_{i}(v)) ≤ K for all i = 1, 2, ⋯, n and all vertices v. Indeed, we can take K = max d_{wd}(∗, g_{i}(∗)), for then

d_{wd}(v, g_{i}(v)) = d_{wd}(h(∗), g_{i}h(∗)) = d_{wd}(∗, h^{−1}g_{i}h(∗)) = d_{wd}(∗, g_{j}(∗))

where h is chosen so that h(∗) = v. It now follows from Proposition 4.12 that each g_{i} is central.

The following corollary answers a question of Jim Carlson. It motivated the construction of X(G) and the analysis of its geometric properties.

Corollary 4.14. Assume that the associated Coxeter group W is irreducible. Let A be a normal abelian subgroup of G. Then A is trivial or < Δ > (and in the latter case Δ is central).

Proof. G acts on A by conjugation. If A is not as in the conclusion, then this action has infinite orbits by Theorem 4.7. The abelian group A is finitely generated by Corollary 4.2. The translation length function induces a norm on the free abelian group A/torsion. The induced action of G must preserve this norm and it still has infinite orbits, a contradiction.

5 The space at infinity

We now construct a “space at infinity” of X(G) and examine the basic properties of the action of G. In a joint work with Mark Feighn it will be shown that elements of infinite order have periodic points at infinity. This is to be regarded as the analog of the space of projectivized geodesic measured laminations for the case of mapping class groups (when A is a braid group, G = A/Δ^{2} is a mapping class group).

5.1 Definition

Recall that a normal form without Δ’s is an oriented path in the Charney graph.

Definition 5.1. An admissible itinerary is an infinite directed path X = x_{1} ·x_{2} ·x_{3} ⋯ in the Charney graph.

We denote by Ω the set of all admissible itineraries and topologize it in the usual fashion: two are close if they agree for a long time. Thus Ω is a totally disconnected compact metrizable space, and it is nonempty unless A is trivial or ℤ. If W is irreducible and nonabelian, then Ω is a Cantor set, by Proposition 4.9 plus the observation that there are vertices with at least two outgoing edges (e.g. if a, b ∈ S don’t commute then there are oriented edges from a to both a and ab). We now describe an action of A^{+} on Ω. Let X ∈ Ω be as above and let g ∈ A^{+}. Observe that by Proposition 1.2 y_{1} = α(gx_{1}) = α(gx_{1}x_{2} ⋯ x_{k}) for any k ≥ 1. Similarly, by Proposition 1.4 the second atom y_{2} in the normal form for gx_{1}x_{2} equals the second atom in the normal form for gx_{1}x_{2} ⋯ x_{k} for any k ≥ 2. Continuing in this fashion, we see that the normal forms for gx_{1}x_{2} ⋯ x_{k} “converge” as k → ∞ to an infinite sequence y_{1} · y_{2} · y_{3} ⋯ so that any finite initial piece is a normal form. This may not be an admissible itinerary since y_{i} might be Δ. However, observe that only finitely many y_{i}’s can be Δ. They all occur at the beginning and their number is no larger than the atomnorm of g. If the number of Δ’s is even we erase them, and if the number is odd we erase them and replace all the remaining y_{i} by their “conjugate” \overline{y}_{i}. Intuitively, we think of pushing all the Δ’s off to infinity, much in the same way as we calculate the special representative of a vertex of X(G).

Since Δ^{2} acts trivially, there is an induced action of G on Ω. The action is continuous (this follows from Lemma 1.4).

5.2 Faithfulness and minimality

Proposition 5.2. If A is not 1, ℤ, or ℤ × ℤ, then the kernel of the action of G = A/Δ^{2} on Ω is either trivial or < Δ > (if Δ is central).

Proof. If Δ is not central, then there is a generator a such that \overline{a} = b ≠ a. Then a^{∞} := a·a·a ⋯ ∈ Ω is not fixed by Δ; indeed its image is b^{∞}.

Let g ∈ A^{+} be nontrivial. Assume first that the normal form of g = x_{1} · x_{2} ⋯ x_{k} does not have any Δ’s. Let a be a generator such that x_{k}·a is a normal form (i.e. so that x_{k}a is not an atom – e.g. a can be the last letter in a word representing x_{k}). Then a^{∞} := a·a·a ⋯ ∈ Ω is fixed by g only if g = a^{k} is a power of a. If b is a generator that does not commute with a, then a^{k}(b^{∞}) = a·a ⋯ a·ab·b·b ⋯ and b^{∞} is not fixed by g. Finally, if b is a generator distinct from a and commuting with a, but there are other generators (so that Δ ≠ ab), then a^{k}(b^{∞}) = ab · ab ⋯ ab · b · b ⋯, so b^{∞} is not fixed in that case either.

Finally, assume that the normal form of g is Δ · x_{1} · · · · · x_{k} with k ≥ 1 and x_{1} ≠ Δ. Again take a to be a generator with x_{k} · a a normal form. If g fixes a^{∞}, then \overline{x}_{1} · · · · · \overline{x}_{k} · \overline{a} · \overline{a} ⋯ = a · a ⋯ and so g = Δa^{k} and \overline{a} = a. If b is a generator such that \overline{b} does not commute with a, then Δa^{k}(b^{∞}) = a^{k}\overline{b}^{∞} = a·a·· · ··a·a\overline{b}·\overline{b}·· · · and so b^{∞} is not fixed. Finally, if A is abelian but has more than 2 generators and if b is a generator distinct from a, then Δa^{k}b^{∞} = ab · ab · · · · · ab · b · b · · · ·.

Proposition 5.3. Suppose that W is irreducible and nonabelian. Then each orbit in Ω is dense. In particular, there is no proper closed invariant subset of Ω.

Proof. Let X = x_{1} ·x_{2} ⋯ and Y = y_{1} ·y_{2} ⋯ be two points in Ω. We need to construct some g ∈ A^{+} such that gX and Y agree in the first n slots. By Proposition 4.9 there is a finite directed path y_{n} · z_{1} ⋯ z_{k} · x_{1} from y_{n} to x_{1}. Take g = y_{1}y_{2} ⋯ y_{n}z_{1} ⋯ z_{k}.

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