New York Journal of Mathematics

New York J. Math. 4 (1998) 237–248.

Buildings and Non-positively Curved Polygons of Finite Groups

Paul R. Brown

Abstract. Let P be a non-positively curved polygon of finite groups. The group P acts on a contractible 2-complex X_{P}, and we prove that this complex is a building if and only if the links have (angular) diameter π. When P has zero group theoretic curvature, a geometric argument shows that the periodic apartments are dense in the set of all apartments.

Contents

1. Introduction 237 1.1. Related results 238

2. Polygons of groups 238 3. Links of diameter π 239

3.1. Diameter, girth, and completing partial circuits 239 3.2. Expanding planar disks 241

4. Buildings 243 4.1. Diameter π and buildings 243 4.2. Squares of groups and tree × tree 244

5. Two-dimensional Euclidean buildings and ℤ ⊕ ℤ’s 244 5.1. Flat polygons of finite groups acting on Euclidean buildings 245

References 247

1. Introduction

Theorem 4.2 (below) connects local combinatorial properties (the metric diameter of vertex links) with global geometric properties (being a building) of the developing complex of a non-positively curved polygon of groups. The technique of the proof of Theorem 4.2 is used to prove several results. Theorem 5.1 shows that

Received March 11, 1998. Mathematics Subject Classification. 20F32, 51E24. Key words and phrases. triangle of groups, polygon of groups, nonpositive curvature, Euclidean

building, hyperbolic building, periodic flats.

This work is based on research partly supported by the National Science Foundation under Grant No. DMS-9503034 and initially appeared as a chapter in the author’s dissertation [4].

\begin{matrix}c \\ ◯ \end{matrix}1998 State University of New York ISSN 1076-9803/98

for a non-positively curved polygon G of finite groups acting on a two-dimensional Euclidean building B, a positive power of almost every element of infinite order in G lies in a ℤ ⊕ ℤ subgroup of G. It follows (Corollary 5.3) that the set of periodic apartments is dense in the set of all apartments in the sense that any flat in B intersects some flat stabilized by a ℤ ⊕ ℤ < G in a ball of arbitrarily large radius.

1.1. Related results. Presenting these results at conferences, I was kindly directed to a number of related results in the literature. Roger Alperin and Jon Corson directed me to two papers of Jacque Tits [13, 12]. By different means and in more general circumstances, Theorem 4.2 was first proved by Tits; Tits’s techniques are combinatorial in nature. Frédéric Paulin brought to my attention a paper of Sylvain Barré [1] where he gives a similar characterization of buildings in two dimensions.

2. Polygons of groups

A polygon of groups or k-gon of groups P is the colimit of a commutative diagram of groups and proper inclusions modelled on the poset of faces of the first barycentric subdivision of a polygonal 2-cell with k ≥ 3 sides. For instance, a square of groups is the colimit of a diagram of the form

P = colim (\begin{matrix}V_{1} & ⟵ & E_{1,2} & ⟶ & V_{2} \\ ↑ & & ↑ & & ↑ \\ E_{4,1} & ⟵ & F & ⟶ & E_{2,3} \\ ↓ & & ↓ & & ↓ \\ V_{4} & ⟵ & E_{3,4} & ⟶ & V_{3} \end{matrix})

It is a convenient abuse of notation to confuse the diagram and the colimit, and for the purposes of this paper, such abuse is harmless. The V_{i} are vertex groups, the E_{i,j} are edge groups, and F is the face group. The diagram defines projections

p_{j} : E_{i,j} ∗_{F} E_{j,k} ⟶ V_{j}.

If p_{j} is not injective, let w_{j} be a shortest non-trivial word in ker p_{j}. The angle at V_{j} is defined to be 2π/|w_{j}| if p_{j} is not injective and 0 if p_{j} is injective.

Let G be a group and {G_{α}} be a set of subgroups. The coset complex of G relative to {G_{α}} is the nerve of the covering of G by cosets of the G_{α}. The canonical example of a coset complex is the tree for a free product of non-trivial finite groups: the tree upon which F = G ∗ H acts is the coset complex for F relative to {G, H}. If the vertex groups of a polygon of groups inject into the colimit, the polygon of groups is developable. The developing complex X_{P} for a developable polygon of groups is the coset complex of P relative to the set of vertex groups, i.e., the combinatorial cell complex whose vertices are the left cosets of P modulo the vertex groups, whose edges are the left cosets of P modulo the edge groups, and whose faces are the elements of P/F; the incidence relation is non-empty intersection, and P acts on the complex by multiplication.

Angles can also be (equivalently) defined combinatorially in terms of the coset complex of V_{j} relative to {E_{i,j}, E_{j,k}}; this coset graph is simply a quotient of the tree on which E_{i,j} ∗_{F} E_{j,k} acts. If p_{j} fails to be injective, the angle at V_{j} is 2π divided by the girth^{1} of the coset graph. If p_{j} is injective, the angle is defined to be zero. Supposing that P is developable, the coset graph of V_{j} relative to {E_{i,j}, E_{j,k}} is the link of a vertex corresponding to a coset of V_{j}. With this in mind, the link of a pair of distinct proper subgroups H, K < G, abbreviated Lk (H ↪ G ↩ K), is the coset complex for G relative to {H, K}. When the angle is non-zero, the link carries a natural metric such that the girth of the graph is 2π.

A k-gon of groups is non-positively curved if the sum of the angles at the vertex groups is no more than (k − 2)π; note that the definition of polygon of groups guarantees that the angle at a vertex of a polygon of groups is no larger than π/2, so every k-gon of groups for k ≥ 4 is non-positively curved. Gersten and Stallings [11] proved:

Theorem 2.1 (Gersten-Stallings). Let P be a non-positively curved polygon of groups. Then,

1. P is developable.

2. The developing complex X_{P} is contractible and carries a CAT(0) or CAT(−1) metric.

The metric on X_{P} is obtained by equivariantly metrizing each 2-cell as though it were a polygon in 𝔼^{2} or ℍ^{2} with geodesic boundary and geometric angles corresponding to the angles at the vertex groups of P.

3. Links of diameter π

The first essential ingredient is a characterization of vertex links whose metric diameter is π.

3.1. Diameter, girth, and completing partial circuits. Let Γ be a finite, connected bipartite graph with no vertices of valence one; the bipartition implies that the girth ℓ of Γ is even. Use a path metric that assigns each edge equal length to metrize Γ so that Γ has metric girth 2π. The angle associated to Γ, ∠Γ = 2π/ℓ is well-defined (namely the length of an edge). The diameter of Γ, denoted diam(Γ) is the maximum distance between any two vertices. Note that diam(Γ) ≥ π, because the shortest circuit in Γ has length 2π.

The situation where Γ has (minimal) diameter π can be characterized in terms of (simplicial) rays of length π + ∠Γ.

Proposition 3.1. Let Γ be a finite, connected bipartite graph with with no vertices of valence one. Then, diam(Γ) = π if and only if every simplicial ray α of length π + ∠Γ is part of a unique circuit of length 2π.

Proof. Suppose that the diameter of Γ is π. Then, there is a geodesic segment β of length less than or equal to π connecting the endpoints of α. If the length of β is π, then the circuit α ∪ β contains an odd number of edges (which is impossible in a bipartite graph). If β is shorter than π − ∠Γ, then the circuit α ∪ β has length shorter than 2π, contradicting the assumed girth of Γ. Thus, the length of β is π − ∠Γ as claimed. Conversely, if Γ contains a geodesic of length strictly larger than π, then that geodesic is not part of any circuit of length 2π.

1The girth of a graph is the length of the shortest non-trivial circuit.

Figure 1. The link of (⟨a⟩ ↪ ⟨a, b : a^{3}, abā = b^{2}⟩ ↩ ⟨ab⟩) has
angle \frac{π}{3} and diameter π.

To establish uniqueness, observe that the intersection of two circuits γ and γ′ of length 2π is a geodesic segment; otherwise,

(γ ∪ γ′) \ (γ ∩ γ′)

would contain a circuit of length 2π − 2∠Γ. □

Having diameter π has strong consequences for the structure of Γ. For example, a deep theorem of Walter Feit and Graham Higman [6] states that if Γ is finite and the valence of each vertex is at least three, then

diam(Γ) = π ⟹ ∠Γ ∈ \{\frac{π}{2} , \frac{π}{3} , \frac{π}{4} , \frac{π}{6} , \frac{π}{8} \} .

The statement of the theorem also gives algebraic relations between the valences of the two types of vertices. Bipartite graphs with girth 2m, diameter m (each edge has length one), and no vertices of valence one are called generalized m-gons in the literature. See, e.g., Ronan [10, pp. 28–30,36–38].

For present purposes, Γ will be the link of a vertex in the developing complex for some polygon of finite groups, in which case the vertex stabilizer G will act transitively and without inversion on the edges of Γ. It is convenient to be able to check the diameter of Γ with knowledge only of G. The following proposition provides one method.

Proposition 3.2. Let G be a group acting on a finite bipartite graph Γ so that Γ/G is a single edge. Suppose diam(Γ) = π, ∠Γ = \frac{π}{m}, and let the edge e = [u, v] be a fundamental domain for the action of G on Γ. Then, G is a quotient of G_{u} ∗_{G_{e}} G_{v} and:

• G = \overbrace{G_{u}G_{v}G_{u}G_{v} …}^{m factors}.

• The order of G is |G_{e}| times the volume (in edges) of a ball of radius π in the tree on which G_{u} ∗_{G_{e}} G_{v} acts.

Proof. Consider the universal cover Γ̃ of Γ. The diameter of Γ is π and the girth of Γ is 2π, so the number of edges in a ball of radius π (about a vertex) in Γ̃ equals the number of edges in Γ. The first statement follows from Bass-Serre theory, and the second statement follows from the fact that the number of elements in G is |G_{e}| times the number of edges in Γ. □

Corollary 3.3 (Uniqueness of F_{21}). The Frobenius group of order 21,

F_{21} = ⟨x, y : x^{3}, y^{3}, xyx̄ȳxȳ⟩ (1) and ℤ_{3} ⊕ℤ_{3} are the only finite groups generated by two ℤ_{3} subgroups forming a link of diameter π.

Proof. If two distinct ℤ_{3} subgroups form a link of diameter π and angle \frac{π}{2} within a group G, then Proposition 3.2 implies that G has order nine. the cylic group of order nine does not contain two distinct cyclic subgroups of order three, so ℤ_{3} ⊕ ℤ_{3} is the only possiblity. If two distinct subgroups of order 3 form a link of diameter π and angle \frac{π}{3} within a group G, then Proposition 3.2 implies that G has order 21. The Frobenius group of order 21 is the only possibility for G, as the other group of order 21 is abelian and any two subgroups of an abelian group form an angle no smaller than \frac{π}{2} . (The link of (ℤ_{3} ↪ F_{21} ↩ ℤ_{3}) appears in Figure 1.) The Sylow theorems and some straightforward observations are sufficient to rule out groups of orders 45, 189, and 765 (corresponding to the possible angles of π/4, π/6, and π/8) and the Feit-Higman Theorem guarantees that there exist no groups G (such that the link would have diameter π) for angles smaller than \frac{π}{8} ; so F_{21} and ℤ_{3} ⊕ ℤ_{3} are the only finite groups where two ℤ_{3} subgroups can form a link of diameter π. □

If the restriction on the diameter of the link is removed, then the possibilities vary widely. For example, consider the following pair of subgroups of a group of order 360:

(⟨x⟩ ↪ ⟨x, y : x^{3}, y^{3}, xyxyx = yxyxy⟩ ↩ ⟨y⟩) . (2)

The two specified ℤ_{3} subgroups form an angle of \frac{π}{5} , so the corollary implies that the diameter of the corresponding link is greater than π. (The diameter is \frac{11π}{5} .) Equivalently, the order of the group is 360, significantly larger than the 189 required for the link to have diameter π.

3.2. Expanding planar disks. Let P be a non-positively curved polygon of groups such the vertex links all have diameter π. A planar disk (respectively, a plane) D in X_{P} is a subcomplex which is convex, homeomorphic to a closed disk in ℝ^{2} (respectively, to ℝ^{2}) and such that the total angular measure around an interior vertex is 2π. For example, in the case of a square of groups with all angles \frac{π}{2} , a planar disk would be an n × m rectangular grid of squares meeting four per interior vertex. For any subgraph L ⊂ Lk v, the cellular hull of L is the corresponding subcomplex of St v.

The lemma and propositions which follow describe the connection between link diameter π and planar subspaces of the developing complex X_{P}.

Lemma 3.4. Let P be a non-positively curved polygon of groups with links of diameter π. Given any convex planar disk Δ ⊂ X_{P}, there exists a convex planar disk Δ′ ⊃ Δ such that ∂Δ ⊂ int Δ′.

Proof. The idea of the proof is depicted in Figure 2 for the case of a (\frac{π}{3} , \frac{π}{3} , \frac{π}{3} ) Euclidean triangle of groups. Any planar disk Δ is the union of planar neighborhoods of the interior vertices and planar neighborhoods of edges, so enlarging Δ is equivalent to choosing a compatible set of planar neighborhoods of the vertices of ∂Δ.

Set Δ′ = Δ and choose a vertex u ∈ ∂Δ such that the angle of ∂Δ at u is less than π. Let u′ be the next vertex of ∂Δ in counterclockwise order from u and choose a minimal circuit γ of Lk u′ that contains Lk_{Δ}u′; add the cellular hull of γ to Δ′.

Figure 2. “Proof” of Lemma 3.4

Fix an orientation on Δ and let u′′ be the next vertex of ∂Δ in counterclockwise order from u′. The angle of ∂Δ at u′′ was at most π, so Lk_{Δ′}u′′ is a segment of length less than or equal to π + ∠Lk v′′. Choose a compatible minimal circuit γ′ ⊂ Lk u′′ and add the cellular hull of γ′ to Δ′. Proceed in this fashion around ∂Δ until it is necessary to adjoin a planar neighborhood of u. The vertex u was chosen so that the angle of ∂Δ′ at u would be π or less at this stage, so there is a minimal circuit in Lk u containing Lk_{Δ′}u; add the corresponding planar neighborhood to Δ′.

The only possible complication at this point is caused by orphaned vertices as shown in Figure 3. Three adjacent 2-cells around a \frac{π}{2} vertex are not a convex configuration for a link of girth 2π. At each orphaned vertex, there exist unique cells to complete planar neighborhoods of the orphaned vertices, and once these cells are added, an interior angle of ∂Δ′ is no more than twice one of the angles of P. Thus, Δ′ is the desired convex planar disk. (Note that a local embedding of a planar disk into the developing complex must be an embedding, as otherwise there would be a closed geodesic.) □

Figure 3. The disk Δ′ with an orphaned vertex v.

If all vertex links have diameter π, this lemma can be used to isometrically embed a plane in the developing complex by successively expanding an embedded convex cellular disk. If the diameter of one of the links is larger than π, it is possible that the process of “growing” a plane by successive applications of Lemma 3.4 stalls at some point. The following proposition will prove useful later.

Proposition 3.5. Let P be a non-positively curved polygon of finite groups with a vertex link L with diam(L) > π. There exist a pair of 2-cells in X_{P} which can not both lie in a single plane.

Proof. There is a geodesic in L with length π + ∠L, so the initial and terminal edges can not both be on a minimal circuit of L, i.e., the corresponding two 2-cells can not both lie in any planar subset of X_{P}. □

4. Buildings

4.1. Diameter π and buildings. After defining buildings, we prove one of the main theorems.

A Coxeter group is a group with a presentation of the form

⟨x_{1}, …, x_{k} : x_{1}^{2}, …, x_{k}^{2}, (x_{1}x_{2})^{m_{1,2}}, …, (x_{i}x_{j})^{m_{i,j}}, …, (x_{k}x_{k−1})^{m_{k,k−1}}⟩ (3)

subject to the requirements that the m_{i,j} are from a symmetric matrix with coefficients in ℤ^{+} ∪ {∞} and 1’s down the diagonal. When the order of x_{i}x_{j} is “∞”, the relation is omitted.

For any k-gon P in the 2-sphere S^{2}, 𝔼^{2} or ℍ^{2} with interior angles π/α_{1}, …, π/α_{k}, there is an associated Coxeter group

Δ(α) = ⟨x_{1}, …, x_{k} : x_{1}^{2}, …, x_{k}^{2}, (x_{1}x_{2})^{α_{1}}, …, (x_{i}x_{i+1})^{α_{i}}, …, (x_{k}x_{1})^{α_{k}}⟩ , (4)

where α = (α_{1}, …, α_{k}) and α_{i} ≥ 2. In the context of polygons of groups, the Coxeter group Δ(α) is a polygon of finite groups with ℤ_{2} edge groups and dihedral vertex groups, and the Coxeter complex of Δ(α) is X_{Δ(α)}. In the study of twodimensional buildings, only the Coxeter complexes for planar reflection groups are necessary, so the technicalities of a more general definition can be sidestepped. Ken Brown’s book on buildings ([3, ch. III]) contains a general discussion.

Definition 4.1. A cell complex X is a building if

1. X is the union of a set of apartments, i.e., subcomplexes, isomorphic to X_{Δ(α)} for some planar polygon P.

2. Given any pair of cells {σ, τ}, there is an apartment A ⊃ (σ ∪ τ).

3. If σ and τ are two top-dimensional cells of X contained in the intersection of two apartments A and B, then there is a cellular isomorphism A ⟶ B fixing σ and τ.

These conditions guarantee that the apartments of X are all the same Coxeter complex. Spherical buildings and Euclidean buildings (also called affine buildings in the literature) buildings, named for the structure of their underlying Coxeter complexes, are well-known for their applications in the study of Lie groups and algebraic number theory. (See, e.g., Ken Brown’s book [3, ch. VI–VII].) The theory of hyperbolic buildings, i.e., buildings in which each apartment is an ℍ^{n} tessellated by the action of a hyperbolic reflection group, is less well developed. (See, e.g., Gaboriau and Paulin [7].)

Theorem 4.2 (J. Tits [13, 12]). Let P be a non-positively curved polygon of finite groups. The vertex links all have diameter π if and only if X_{P} is a building.

Proof. We verify the conditions in Definition 4.1.

Let {P_{α}} be the set of all planes in X_{P}. We take these to be the apartments of X_{P}. The first condition follows from Lemma 3.4, as any 2-cell is trivially a convex, isometrically embedded cellular disk.

The metric 2-complex X_{P} is geodesic, so the centers of any two 2-cells σ and τ can be connected with a geodesic segment L. The diameter of the link is the minimal π, so if ℓ runs through a vertex, its points of entry and exit form an angle of π. It follows from the ideas of the proof Lemma 3.4 that a convex cellular disk containing σ∪τ ∪ℓ can be created by growing a disk around σ so that each successive expansion contains more of ℓ. This process might require adding a triangle in the middle of a boundary segment of a disk and then tiling along a boundary segment in two directions, but the program of Lemma 3.4 is unaffected: this procedure still tiles-in the boundary segment of the disk in question, and the other boundary segments can be dealt with in counter-clockwise order as in the Lemma. This done, repeated application of Lemma 3.4 provides a family of embedded cellular disks which is strictly increasing with respect to inclusion; the union of this family is the desired plane.

Let P_{i} and P_{j} be such that σ ∪ τ ⊂ P_{i} ∩ P_{j}. The intersection of planar subspaces is planar, so Δ = P_{i} ∩ P_{j} is a possibly unbounded convex planar disk. The planes P_{i} and P_{j} are simply expansions of Δ in the spirit of Lemma 3.4, so the desired isomorphism is simply the limit of the isomorphisms between the incremental expansions of Δ as subsets of P_{i} and P_{j}.

The reverse implication follows from Proposition 3.5. □

4.2. Squares of groups and tree × tree. The following proposition illustrates an application of Theorem 4.2 to squares of groups.

Proposition 4.3. Let S be a non-positively curved 4-gon of finite groups. The following are equivalent:

1. The vertex links of S are connected with diameter π and angle π/2.

2. The vertex links of S are complete bipartite graphs.

3. X_{S} is the product of two locally finite, uniform bipartite trees.

4. The order of each vertex group is the product of the orders of the incident edge groups divided by the order of the face group.

Proof. The product of buildings is again a building, so 3 ⟹ 1 is immediate. The cells of a Euclidean building are either simplices or products of simplices, and when the cells are the product of lower-dimensional simplices, the building decomposes as a product of lower-dimensional buildings. (See [10, p. 110].) In this case, the cells of the building X_{S} are the product of two 1-simplices and X_{S} decomposes as the product of two 1-dimensional Euclidean buildings, i.e., trees. The local finiteness of X_{S} implies that the trees have finite valence, and thus 1 ⟹ 3. The equivalence of 1 and 2 is elementary, and the equivalence of 1 and 4 follows from the order computation in Proposition 3.2. □

The tree×tree developing complexes have a particularly simple topological structure, but the polygons of groups acting on these complexes exhibit widely varied behavior. C. Y. Tang and G. Kim [9] have proven residual finiteness and subgroupseparability results for squares of nilpotent groups with cyclic edge groups and trivial face group, and Dani Wise [14] has produced non-residually finite flat squares of finite p-groups with developing complex tree × tree. Marc Burger and Shahar Mozes [5] have produced finitely presented infinite simple groups acting on tree×tree.

5. Two-dimensional Euclidean buildings and ℤ ⊕ ℤ’s

5.1. Flat polygons of finite groups acting on Euclidean buildings. A polygon of groups is flat if it has the angles of a Eulcidean polygon. Let P be a flat polygon of finite groups with X_{P} a building. To this point, we have been concerned with the existence and number of planes in X_{P}. Using a number of techniques inspired by Lemma 3.4, we now focus on ℤ ⊕ ℤ subgroups of P.

We begin with a variation on the idea of Lemma 3.4. Let w be an element of P. The vertex links of a building are necessarily connected (by condition 2 in Definition 4.1), so the edge groups generate P and w can be expressed as a word in elements of the edge groups in many different ways. Fix an initial 2-cell σ ⊂ X_{P}. A row of 2-cells in X_{P} is the area between two parallel cellular geodesic segments at minimum distance from each other. A zig in X_{P} is the union of two rows where the parallel cellular geodesics meet each other at the maximum angle, e.g., π − \frac{π}{6} for a (\frac{π}{2} , \frac{π}{3} , \frac{π}{6} ) triangle. The cells σ and w ·σ both lie in some plane Q ⊂ X_{P} (condition 2 in Definition 4.1), and σ and w · σ are connected by either a zig or a row lying entirely on Q. The element w is straight if σ and w · σ lie in some row and bent otherwise.

(a) (b)

Figure 4. (a) A zig connecting two cells (shaded darker) for a (\frac{π}{2} , \frac{π}{4} , \frac{π}{4} ) triangle. (b) a partial sequence of zigs for a (\frac{π}{3} , \frac{π}{3} , \frac{π}{3} ) triangle with the w-translates of σ shaded.

A pair of two cells on a plane Q have the same relative orientation if one is the image of the other under a translation acting on the induced tiling of Q and this translation preserves stabilizer types. A nontrivial element w is a translation if w and w ·σ have the same relative orientation in some (and thus any) plane. A pair of 2-cells occupies a finite set of relative orientations, so given an element w of infinite order, there exists a number n such that w^{n} is a translation.

Bent translations have non-trivial centralizers:

Theorem 5.1. Let P be a flat polygon of finite groups with X_{P} a Euclidean building. For any bent element w of infinite order, there exists an element v and n ∈ ℤ^{+} such that ⟨w^{n}, v⟩ ≅ ℤ ⊕ ℤ. Equivalently, any bent translation lies in a ℤ ⊕ ℤ < P stabilizing a plane in X_{P}.

Proof. Suppose that w is a bent translation, fix a zig representing it, a two-cell σ, and consider the subset Z_{0}(w) ⊂ X_{P} which is the union of the zigs connecting w^{i} · σ with w^{i+1} · σ for i ∈ ℤ. The boundary of Z_{0}(w) is composed of cellular geodesics, and the angles at the boundary vertices of Z_{0}(w) are π, π + ϵ, or π − ϵ for ϵ the smallest angle of P. Select a vertex v with angle π + ϵ, and complete the neighorhood of v in Z_{0}(w) to a flat disk by adding 2-cells. Using the ideas of the proof of Lemma 3.4, the consequences of this addition give a new sequence of zigs Z_{1}(w) parallel to Z_{0}(w) as in Figure 5. Observe that Z_{0}(w) and Z_{1}(w) are the same shape by construction.

Figure 5. A part of Z_{0}(w) and Z_{1}(w) for a (\frac{π}{2} , \frac{π}{3} , \frac{π}{6} ) triangle

By selecting two cells in the same relative positions in Z_{1}(w) as σ and w · σ occupy in Z_{0}(w), the new sequence of zigs Z_{1}(w) represents a translation w_{1} ∈ P. The translation w_{1} is determined by w and has the same format and length as a word in the generators of the edge groups of P.

In the same fashion, Z_{1}(w) uniquely determines a sequence of zigs Z_{2}(w) with an associated word w_{2} and so on, giving a sequence Z_{i}(w) and words w_{i} for i ∈ ℤ^{+}. By a mathematical headstand, the process is valid for negative indices as well. The assumed finiteness of the edge groups implies that there are only finitely many words of the same length and format as w. Each element is determined by the preceding element (or equivalently, the subsequent element) in the bi-infinite sequence …, w_{−1}, w, w_{1}, …, so every element which occurs once occurs infinitely often.

Because w occurs twice, say as w_{0} and w_{n}, the uniform periodicity of the collections of zigs gives a “vertical” word v such that wvw^{−1}v^{−1} fixes σ.

An elementary lemma completes the proof.

Lemma 5.2. Let x and y be two elements of infinite order in a group G and suppose that all of the elements {[x, y^{k}] : k ∈ ℤ^{+}} lie in a finite subset of G. Then, there exists n ∈ ℤ^{+} such that [x, y^{n}] = 1.

Proof. Infinitely many [x, y^{n}] lie in a finite subset of G, so some two must be equal, say [x, y^{j}] = [x, y^{k}] with j ≠ k. Then,

\begin{aligned}xy^{j}x̄ȳ^{j} & = xy^{k}x̄ȳ^{k} \\ y^{j−k}x̄y^{k−j} & = x̄. \end{aligned} (5)

and [x, y^{k−j}] = 1 as desired. □

The proof of Theorem 5.1 is complete. □

A plane in X_{P} is periodic if it is stabilized by a ℤ ⊕ ℤ subgroup of P.

Corollary 5.3. For a flat polygon of finite groups P with links of diameter π, any planar disk in X_{P} is contained in a periodic plane, i.e., the set of periodic planes is a system of apartments for X_{P}.

Buildings and Polygons of Finite Groups 247

Proof. Given any planar disk Δ ⊂ X_{P}, the disk can be enlarged until it is a parallelogram with a pair of zigs for boundaries; this is equivalent to the parallelogram having the sharpest possible angles at two corners. Carrying out the construction of Theorem 5.1 with either of the associated words (which both represent the same element) gives a plane Q ⊃ Δ with a ℤ ⊕ ℤ subgroup of P stabilizing Q. □

Equivalently,

Corollary 5.4. Every plane in X_{P} is the limit of a sequence of periodic planes.

Proof. Let Q be a plane in X_{P} and let {Δ_{i}} be a sequence of compact, convex cellular subsets of Q with Δ_{i} ⊂ Δ_{i+1} and Q = ⋃ Δ_{i}. Corollary 5.3 implies that there exists a sequence {Q_{i}} of periodic planes with Δ_{i} ⊂ (Q_{i} ∩ Q) and

⋃ (Q_{i} ∩ Q) = Q.
□

It would be interesting to know whether similar results hold for hyperbolic buildings:

Question 5.5. In a cocompact 2-dimensional hyperbolic building, is the set of periodic (under the actions of hyperbolic surface groups) apartments dense? Do the periodic apartments form a system of apartments for the building?

References

[1] Sylvain Barré, Polyèdres finis de dimension 2 à courbure ≤ 0 et de rang 2, Annales L’Institut Fourier, Grenoble 45 (1995), no. 4, 1037–1059, MR 96k:53056.

[2] Martin R. Bridson, Geodesics and curvature in metric simplicial complexes, Group Theory from a Geometrical Viewpoint (Etienne Ghys, André Haefliger, and Alberto Verjovsky, eds.), World Scientific, River Edge, NJ, 1991, pp. 373–463, MR 94c:57040.

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Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, USA

Current address: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607

prb@uic.edu http://klee.math.uic.edu/˜ pbrown/

This paper is available via http://nyjm.albany.edu:8000/j/1998/4-15.html.