Combinatorial structure of some hyperbolic buildings.

Alina Vdovina

Department of Mathematical Sciences

SUNY, Binghamton, NY, 13 902

e-mail: vdovina@math.binghamton.edu

Abstract

We construct polyhedra whose links are combinatorial one-dimensional buildings. The polyhedra of this type are interesting because of their universal coverings, which are two-dimensional buildings.

Introduction

We will call a polyhedron a two-dimensional complex, which is obtained from several oriented p-gons by identification of corresponding sides. Let’s consider a point of the polyhedron. Let’s take a sphere of a small radius at this point. The intersection of the sphere with the polyhedron is a graph, which is called the link at this point.

We consider the polyhedra with the following properties: all links of all vertices are combinatorial one-dimensional buildings.

Definition. A combinatorial one-dimensional building such that it consists of its edges called chambers and a family of subgraphs called apartments such that:

1. Every apartment is a cycle composed of 2N edges for some fixed N;

2. For any two chambers there is an apartment containing both of them;

3. If two apartments have a common chamber, so there is an isomorphism of these apartments fixing their intersection pointwise.

Definition. Let P(k, m) a tessalation of the hyperbolic plane by a regular polygon with k sides (k ≥ 5), with angles π/m in each vertex (with m entier, m ≥ 2). We will call hyperbolic building a polygonal complex X, which can be expressed to the union of subcomplexes isomorphic to P(k, m) called apartments satisfying the following axioms:

(i) each point of X is an apartment;

(ii) for any two polygons of X, there is an apartment containing both of them;

(iii) for any two apartments A_{1}, A_{2} ∈ X containing the same polygon, there exists an isomorphism A_{1} → A_{2} fixing A_{1} ∩ A_{2}.

Let C_{p}, p ≥ 5 be a polyhedron whose faces are p-gons and whose links are one-dimensional buildings. We equip every face of C_{p} with the hyperbolic metric, such that all sides of the polygons are geodesics and all angles are π/3, sides which are identified have the same length. Then the universal covering of such polyhedron is a hyperbolic building, see [1],[6].

In the case p = 3, i.e. C_{p} is a simplex, we can also give an euclidean metric to every face. In this metric sides which are identified also have the same length and all sides of the triangles are geodesics. The universal coverings of these polyhedra with euclidean metric are euclidean buildings, see [2], [3].

The main result of our paper is the following theorem. We prove the existence of a polyhedron with a given even number of sides of every face, whose links are one-dimensional combinatorial buildings in every vertex.

Theorem. For every k, q, r there exists a polyhedron with 2k vertices, 2k-gonal faces and one-dimensional spherical buildings on q+r vertices as links in all vertices.

It is known (see [7], for example), that combinatorial buildings are exactly generalized m-gons. A generalized m-gon is a connected, bipartite graph of diameter m and girth 2m, in which each vertex lies on at least two edges. A graph is bipartite if its set of vertices can be partitioned into two disjoint subsets such that no two vertices in the same subset lie on a common edge; the diameter is the maximum distance between two vertices and the girth is the length of a shortest circuit. The vertices of the one subset we will call black vertices and the vertices of the other subset the white ones.

Let G be a generalized m-gon on q + r vertices, q black vertices and r white ones. Let A and B be two alphabets on q and r letters respectively, A = {x_{1}, x_{2}, …, x_{q}} and B = {y_{1}, y_{2}, …, y_{r}}. We mark every black vertex with an element from A and every white vertex with an element from B.

We will also define an incidence matrix of such graph in the following way: the first element of each line is a white vertex and all other elements are black vertices incident to this white vertex. Different lines correspond to different white vertices.

Example 1. For the smallest complete bipartite graph (generalized 2-gon) with four vertices, two black x_{1}, x_{2} and two white y_{1}, y_{2}, the incidence matrix will be

y_{1}: x_{1} x_{2}

y_{2}: x_{1} x_{2}

Let’s define a graph G′, which is inverse to G, by the following way:

Graph G′ has r white vetices, labelled by elements of the alphabet B, and q black vertices, labelled by elements of the alphabet A and vertices y_{i} and x_{j} are incident in G′ in and only if x_{j} and y_{i} are incident in G. Remark. The graph G′ can be obtained from G by changing black and white colors of vertices.

(It is easy to see, that G and G′ are isomorphic as usual graphs.) Similarly, we can define the incidence matrix by ”inverse” way, as correspondence of black vertices to white ones. For the graph from the example we have

x_{1}: y_{1} y_{2}

x_{2}: y_{1} y_{2}

Polygonal presentation.

Definition. Suppose we have n disjoint generalized m-gons G_{1}, G_{2}, … G_{n}. Let P_{i} and L_{i} be the sets of black and white vertices respectively in G_{i}; let P = ∪P_{i}, L = ∪L_{i}, P_{i} ∩P_{j} = ∅ for i ≠ j and let λ be a bijection λ : P → L.

A set K of k-tuples (x_{1}, x_{2}, …, x_{k}), x_{i} ∈ P, will be called a polygonal presentation over P compatible with λ if

(1) given x_{1}, x_{2} ∈ P, then (x_{1}, x_{2}, x_{3}, …, x_{k}) ∈ K for some x_{3}, …, x_{k} if and only if x_{2} and λ(x_{1}) are incident in some G_{i};

(2) (x_{1}, x_{2}, x_{3}, …, x_{k}) ∈ K implies that (x_{2}, x_{3}, …, x_{k}, x_{1}) ∈ K; (3) given x_{1}, x_{2} ∈ P, then (x_{1}, x_{2}, x_{3}, …, x_{k}) ∈ K for at most one

x_{3} ∈ P.

We will call λ a basic bijection.

Polyhedral presentations for n = 1 and k = 3, q = 2, 3 were listed in [5]. They were called triangle presentations in that paper, because the case k = 3 was considered.

Example 2. Let’s take a triangle presentation T_{A_{1}} from [5], p.160:

(0, 1, 3), (1, 2, 4), (2, 3, 5), (3, 4, 6), (4, 5, 0), (5, 6, 1), (6, 0, 2).

We can express it as an incidence matrix of finite projective plane of order two, which corresponds to this polyhedral presentation:

0: 1 4 2

1: 3 2 5

2: 4 3 6

3: 0 4 5

4: 1 5 6

5: 0 2 6

6: 0 1 3

Construction of polyhedra.

We can associate a polyhedron K on n vertices with each polygonal presentation K as follows: for every cyclic k-tuple (x_{1}, x_{2}, x_{3}, …, x_{k}) from the definition we take an oriented k-gon on the boundary of which the word x_{1}x_{2}x_{3} … x_{k} is written. To obtain the polyhedron we identify the corresponded sides of our polygons respecting orientation. We will say that the polyhedron K corresponds to the polygonal presentation K.

Lemma. A polyhedron K which corresponds to a polygonal presentation K has graphs G_{1}, G_{2}, …, G_{n} as the links.

Proof. Let’s take an oriented k-gon for each cyclic k-tuple and write the k-tuple on the boundary of this k-gon. Then we identify the sides with the same label respecting orientation to obtain the polyhedron K. Each face of the polyhedron comes from one k-gon and vertices of the polyhedron come from vertices of the polygons. Look at the face of K which came from the k-tuple (x_{1}, x_{2}, x_{3}, …, x_{k}) and take a vertex v of the polyhedron, such that the edge x_{1} is entering to this vertex v and x_{2} is leaving v. Let’s take a sphere of a small radius with the center in v. The intersection of this sphere with x_{1} will be a white vertex x_{1}, the intersection of this sphere with x_{2} will be a black vertex x_{2} and the intersection of this sphere with the face will be an edge joining x_{1} and x_{2}. But by the definition of the polygonal presentation in the graph G_{1} there is an edge joining the white vertex x_{1} and black vertex x_{2}. Look at all edges, incident to v. On edges entering v we will put white vertices and on edges leaving v we will put black vertices. By the definition of the polygonal presentation there is an edge joining two of these vertices if and only if there is a face, where they are neighbors, and they are neighbors in a face if and only if they are neighbors in some k-tuple.

Let’s look at the incidence matrix of the graph, which is a link in a vertex v of the polyhedron. It follows from the definition of the polygonal presentation, that a graph with such incidence matrix has to be G_{i}.

Proof of Theorem. By the Lemma, to construct the polyhedron with given links, it is sufficient to construct corresponding polygonal presentation.

Let’s take 2k alphabets, A_{i}, i = 1, …, 2k , each of them with i odd consisting of q letters, A_{i} = {x_{1}^{i}, …, x_{q}^{i}} and each of them with i even consisting of r letters, A_{i} = {x_{1}^{i}, …, x_{r}^{i}}.

Let G_{1} be a generalized m-gon on q + r vertices, q white and r black ones. All white vertices of G_{1} will be marked by the elements of the alphabet A_{1} and all black ones of G_{1} will be marked by the elements of A_{2}.

Now we define ordered sets of letters and we will show, that they determine a desired polygonal presentation K:

a 2k-tuple (x_{j}^{1}, x_{s}^{2}, x_{j}^{3}, x_{s}^{4}, …, x_{j}^{2k−1}, x_{s}^{2k}) belongs to K if and only if x_{j}^{1} and x_{s}^{2} are incident in G_{1}, where j = 1, …, q, s = 1, …, r, each 2k-tuple is considered with all its cyclic permutations.

Graph G_{i}, where i is odd, 3 ≤ i ≤ 2k − 1 is defined as follows: all white vertices of G_{i} will be marked by the elements of the alphabet A_{i} and all black ones of G_{i} will be marked by the elements of A_{i+1(mod2k)}; vertices x_{p}^{i} and x_{t}^{i+1} are incident in G_{i} if and only if x_{p}^{1} and x_{t}^{2} are incident in G_{1}.

Graph G_{i}, where i is even 2 ≤ i ≤ 2k, is defined as follows: all white vertices of G_{i} will be marked by the elements of the alphabet A_{i} and all black ones of G_{i} will be marked by the elements of A_{i+1(mod2k)};vertices x_{p}^{i} and x_{t}^{i+1} are incident in G_{i} if and only if x_{t}^{1} and x_{p}^{2} are incident in G_{1}. (Here and below all upper indexes will be taken modulo 2k.)

The basic bijection λ between white and black vertices of all graphs G_{i}, i = 1, …, 2k will be defined as bijection, induced by letters of alphabets A_{i}, that is the image of each black vertex x_{j} is the white vertex, labelled by the same letter x_{j}.

It is necessary to check all axioms of the polygonal presentation for an arbitrary 2k-tuple:

(1) Let’s consider an arbitrary 2k-tuple of K

(x_{n}^{t}, x_{p}^{t+1}, …, x_{l}^{t+2k−1(mod2k)}, x_{h}^{t+2k(mod2k)}).

If t is odd, then λ(x_{n}^{t}) is a white vertex x_{n}^{t} of G_{t}, x_{p}^{t+1} is a black vertex of G_{t} and x_{n}^{t} and x_{n}^{t+1} are incident in G_{t} by constructuion of K. Indeed, for t odd the letter x_{p}^{t+1} follows after x_{n}^{t} if and only if the white vertex x_{n}^{1} and the black one x_{p}^{2} are incident in G_{1}. And vertices x_{n}^{t} and x_{n}^{t+1} are incident in G_{t} if and only if x_{n}^{1} and x_{p}^{2} are incident in G_{1}.

By similar arguments G_{t} for t even is isomorphic to the graph G_{t} which is inverse to G_{1}. But the graph, which is inverse to a generelized m-gon is also a generalized m-gon. So, the first property of a polygonal presentation is already checked.

(2) In our construction we take all cyclic permutations of each 2k-tuple.

(3) Each 2k-tuple is defined by a pair (j, s) in a unic way, where j is the index of letters of alphabets with odd numbers, and s is the index of letters of alphabets with even numbers.

So, K is a polygonal presentation, and by Lemma, it defines a polyhedron on 2k vertices with one-dimensional combinatorial buildings as links in all vertices.

Theorem is proved.

Example 3. Let’s construct a polyhedron which faces are hexagons and a complete bipartite graph with two white and three black vertices as a link in each vertex of the polyhedron. We take six alphabets A_{i}, i = 1, ..., 6, each of them with i odd consisting of two letters, A_{i} = {x_{1}^{i}, x_{2}^{i}} and each of them with i even consisting of three letters, A_{i} = {x_{1}^{i}, x_{2}^{i}, x_{3}^{i}}.

The incidence matrix of G_{i} with i odd :

x_{1}^{i} : x_{1}^{i+1}x_{2}^{i+1}x_{3}^{i+1}

x_{2}^{i} : x_{1}^{i+1}x_{2}^{i+1}x_{3}^{i+1} The incidence matrix of G_{i} with i even : x_{1}^{i} : x_{1}^{i+1}x_{2}^{i+1} x_{2}^{i} : x_{1}^{i+1}x_{2}^{i+1} x_{3}^{i} : x_{1}^{i+1}x_{2}^{i+1} The polyhedral presentation:

(x_{1}^{1}, x_{1}^{2}, x_{1}^{3}, x_{1}^{4}, x_{1}^{5}, x_{1}^{6}), (x_{1}^{1}, x_{2}^{2}, x_{1}^{3}, x_{2}^{4}, x_{1}^{5}, x_{2}^{6}), (x_{1}^{1}, x_{3}^{2}, x_{1}^{3}, x_{3}^{4}, x_{1}^{5}, x_{3}^{6}),

(x_{2}^{1}, x_{1}^{2}, x_{2}^{3}, x_{1}^{4}, x_{2}^{5}, x_{1}^{6}), (x_{2}^{1}, x_{2}^{2}, x_{2}^{3}, x_{2}^{4}, x_{2}^{5}, x_{2}^{6}), (x_{2}^{1}, x_{3}^{2}, x_{2}^{3}, x_{3}^{4}, x_{2}^{5}, x_{3}^{6}).

Each 6-tuple is considered with all its cyclic permutations.

Acknowledgments. I would like to thank Etienne Ghys for his advice to look at this subject and for the helpful discussions. I also thank Werner Ballmann for many consultations concerning euclidean and hyperbiloc buildings. This paper was written during my stay at the Max-Planck-Institut für Mathematik (MPIM), Bonn. We would like to thank the Directors and the staff of the MPIM for hospitality and for providing the excellent working conditions.

References

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