Groups Acting Simply Transitively on Hyperbolic Triangular Buildings
Abstract
We construct and classify all groups, given by triangular presentations associated to the smallest thick generalized quadrangle, that act simply transitively on the vertices of hyperbolic triangular buildings of the smallest nontrivial thickness. Our classification shows 23 nonisomorphic torsion free groups (obtained in an earlier work) and 168 nonisomorphic torsion groups acting on one of two possible buildings with the smallest thick generalized quadrangle as the link of each vertex. In analogy with the Euclidean case, we find both torsion and torsion free groups acting on the same building.
1 Introduction
Intensive study of groups acting simply transitively on Euclidean buildings was initiated in [8] and [9]. This work has had a considerable impact in several directions. For example, this led to new examples of fake projective planes ([16]), and, finally, to their full classification ([10]). In the Euclidean case, there are two nonisomorphic buildings of minimal nontrivial thickness admitting a simply transitive action, and eight isomorphism classes of groups acting on these two buildings simply transitively and in a type preserving manner. Within this isomorphism class, five groups are torsion free and three have torsion [9].
In this paper we study groups acting simply transitively on hyperbolic buildings with the smallest thick generalized quadrangle as the link of each vertex. The torsion free groups acting simply transitively on such buildings were classified in [15]. Here we classify triangle presentations associated to the smallest thick generalized quadrangle, as well as groups with torsion coming from these presentations.
It is known ([17]) that up to isomorphism, there are only two possible triangular hyperbolic buildings, with the smallest generalized quadrangle as the link of each vertex, admitting a simply transitive action. We note that in the formulation of the main theorem in [17] the appropriate polygonal complexes are required to be symmetric, but the proof works also for buildings admitting simply transitive actions.
In [15] the authors constructed, for any , torsion free groups acting cocompactly on hyperbolic buildings with gonal chambers. Our strategy in this paper is to modify the construction in [15] to include the torsion case as well.
Our classification shows 168 nonisomorphic torsion groups acting on one of two possible buildings with the smallest thick generalized quadrangle as the link of each vertex. In analogy with the Euclidean case, we find both torsion and torsion free groups acting on the same building. These groups are listed in Appendix 1. The two possible buildings are denoted by (1) and (2) in Appendix 1.
The link of order 2 for a KacMoody building with the minimal generalized quadrangle as the link of each vertex and equilateral triangular chambers was computed in unpublished paper by the first author and D. Cartwright and T. Steger ([5]), using an invariant for links of order 2 developed by T. Steger. The KacMoody building coincides with our building with number (2).
By [17] there are only two possible isomorphism classes of buildings with the smallest thick generalized quadrangle as the link of each vertex and by results of the present paper at least two of them are nonisomorphic. Thus all the groups from Appendix 1 with building number (2) are cocompact lattices in the automorphism group of the corresponding KacMoody building. It remains to determine if it is possible to embed these lattices into the corresponding KacMoody group.
The existence of cocompact lattices in certain KacMoody groups has already been established. In [7], the authors generalized Lubotzky’s construction of Schottky groups of automorphisms in over a nonarchimedean local field to give torsion free cocompact lattices in any rank 2 locally compact KacMoody group over a finite field . In [4] Capdebosq and Thomas classified cocompact lattices with torsion and with quotient a simplex in rank 2 KacMoody groups corresponding to symmetric generalized Cartan matrices. In [6], the first author and Cobbs showed that over the field with 2 elements, rank 3 KacMoody groups of noncompact hyperbolic type whose Weyl groups are a free product of ’s contain a cocompact lattice that also acts discretely and cocompactly on a simplicial tree. In [2] and [3], Bourdon constructed a family of cocompact lattices in the automorphism groups of certain hyperbolic KacMoody buildings. In [19], Rémy and Ronan showed that Bourdon’s cocompact lattices , , can be embedded into the closure of rightangled KacMoody groups in the automorphism groups of their buildings, for a prime power.
In all of the above cases, the KacMoody buildings are rightangled. The groups we construct here are the first examples of cocompact lattices acting on buildings that are not rightangled.
By [22] it is known that groups acting cocompactly on hyperbolic buildings, in such a way that the chamber is a polygon with at least four sides, are residually finite. But whether or not groups acting cocompactly on triangular hyperbolic buildings are residually finite remains an open question. Our hyperbolic groups acting simply transitively on triangular hyperbolic buildings are possible candidates of such groups that are not residually finite. The commutator subgroups of many of our examples are perfect groups (that is, they have trivial abelianizations) and an extensive computer search (which was carried out since the paper [15] was completed) did not find any normal subgroups of these commutator subgroups.
To prove our main theorem, we used a program written in Fortran to determine the equivalence classes of triangular presentations. We used Magma to determine isomorphism classes of dual graphs of polyhedra and hence of triangle presentations.
2 Definitions and main results
Recall that a generalized gon is a connected, bipartite graph of diameter and girth (the length of shortest circuit) , in which each vertex lies on at least two edges.
We will call a polyhedron a twodimensional complex which is obtained from several oriented gons (Euclidean or hyperbolic) with words on the boundary, by identification of sides with the same labels respecting orientation. We assume that each side of our polygons has length 1.
Consider a sphere of a radius at a vertex of the polyhedron. The intersection of the sphere with the polyhedron is a graph, which is called the link at this point. Consider now a sphere of a radius , at a vertex of the polyhedron. The intersection of this sphere and the polyhedron will be called a link of order two.
We will use the definition of a hyperbolic building given in [14], where an infinite series of examples of hyperbolic buildings, with prescribed local structure, were constructed and studied.
Definition 2.1.
Let be a tessellation of the hyperbolic plane by regular polygons with sides, with angles in each vertex where is an integer. A hyperbolic building is a polygonal complex , which can be expressed as the union of subcomplexes called apartments such that:

Every apartment is isomorphic to .

For any two polygons of , there is an apartment containing both of them.

For any two apartments containing the same polygon, there exists an isomorphism fixing .
Let be a polyhedron whose faces are gons and whose links are generalized gons with . We equip every face of with the hyperbolic metric such that all sides of the polygons are geodesics and all angles are . Then the universal covering of such a polyhedron is a hyperbolic building (see [12]).
Therefore to construct hyperbolic buildings with cocompact group actions, it is sufficient to construct finite polyhedra with appropriate links.
We recall also the definition of a polygonal presentation introduced in [21]:
Definition 2.2.
Suppose we have disjoint connected bipartite graphs . Let and be the sets of black and white vertices respectively in , ; let , , , for and let be a bijection .
A set of tuples , , will be called a polygonal presentation over compatible with if

implies that ;

given , then for some if and only if and are incident in some ;

given , then for at most one .
If there exists such , we will call a basic bijection.
Remark 1. The polygonal presentations with , , and is the smallest generalized 3gon have been listed in [8] and [11].
We use the following definition of equivalence, which is similar to the one in [9].
Definition 2.3.
Let and be two polygonal presentations with , , and for which the graph is a generalized 4gon. Then and are equivalent if there exists an automorphism of the generalized 4gon which transforms the 4gon of to the 4gon of .
Here we classify all polygonal presentations for , and is the smallest thick generalized quadrangle (4gon). Figure 1 shows the graph .
In [15] the authors classified all polygonal presentations for the case , and is the smallest thick generalized quadrangle, when at least two labels in each triangle are different. This corresponds to the case of torsion free groups acting simply transitively on the building.
Theorem 2.4.
([15]) There are 45 nonequivalent torsion free triangle presentations associated to the smallest thick generalized quadrangle. These give rise to 23 nonisomorphic torsion free groups, acting simply transitively on triangular hyperbolic buildings of smallest nontrivial thickness.
It turns out that if we allow torsion in the groups acting simply transitively on hyperbolic triangular buildings, the number of nonequivalent presentations and the number of nonisomorphic groups is much larger:
Theorem 2.5.
There are 7159 nonequivalent triangle presentations corresponding to groups with torsion associated to the smallest generalized quadrangle. These give rise to 168 nonisomorphic groups, acting on one of two possible triangular hyperbolic buildings with the smallest thick generalized quadrangle as the link of each vertex (listed in Appendix 1).
We can associate a polyhedron on vertices with each polygonal presentation as follows: for every cyclic tuple we take an oriented gon on the boundary of which the word is written. To obtain the polyhedron we identify the corresponding sides of the polygons, respecting orientation. We say that the polyhedron corresponds to the polygonal presentation .
The following lemma was proved in [21]:
Lemma 2.6.
A polyhedron which corresponds to a polygonal presentation has graphs as vertexlinks.
Polyhedra corresponding to polygonal presentations from Theorem 2.4 have generalized 4gons as vertexlinks and regular hyperbolic triangles with angles as faces. The universal covering of such a polyhedron is a hyperbolic building (see [12]). Moreover, with the metric introduced in [1, p. 165] this building is a complete metric space of nonpositive curvature in the sense of Alexandrov and Busemann [13]. Examples of hyperbolic buildings with rightangled triangles were constructed by M. Bourdon in [2] and in [12].
Remark. If we have a group with torsion, to apply [12], we have to consider index 3 subgroups (obtained in a canonical way by changing alphabets), to form polyhedra with three vertices, then to go to the universal cover carrying the labels, and then remove the indices of labels.
3 Proof of Theorem 2.5
We construct all polygonal presentations with and and for which the graph is a generalized 4gon. The 23 torsion free groups were listed in [15]. Here we give the groups with torsion. Our strategy is to go through all possible incidence tableaus for and determine if they can be interpreted as triangle presentations.
Let be the set of black vertices and be the set of white vertices in . We denote the elements of by and the elements of by , . In all cases we define the basic bijection by .
By [20], the smallest thick generalized 4gon can be presented in the following way: its “points” are pairs , where , and “lines” are triples of those pairs, where and are all different. Therefore, we build a tableau as follows: For each row we take three pairs , and , where and are all different and in . These are our points: , ,…, .
Next we label the rows in Table 1 by in such a way that the result is an incidence tableau that gives a triangle presentation with the basic bijection . To obtain groups with torsion, we demand that at least one of the triangles is of the form . For example, labeling the rows from top to bottom by , , , , , , , , , , , , , , and gives rise to the presentation with the following 17 triangles: , , , , , , , , , , , , , , , and .
The labeling of rows in Table 1 defines the triangles uniquely: since the last row , , has label we know that there are triangles , and for some points , and . For the first of these triangles the missing point is , since the line has points , and and from the lines with those respective numbers only has the point . That is, line has point , line has point and line has point , and this gives the triangle . Similarly, we must have and . Going through all the rows we get the triangles for this presentation. The number of the triangles in each presentation is either 17 or 19, depending of whether there is 3 or 6 triangles of the from .
The presentations are searched by a computer program. The program is written in Fortran in order to keep it fast and simple. It goes through all 15! ways to label the rows of the given tableau, and decides, which of these give an incidence tableau of a triangle presentation with torsion. The program outputs one representative of each equivalence class of triangle presentations. We obtain in this way 7159 different equivalence classes of presentations.
For a polygonal presentation , take ( or ) oriented regular hyperbolic triangles with angles , write words from the presentation on their boundaries and glue together sides with the same letters, respecting orientation. The result is a hyperbolic polyhedron with one vertex and triangular faces, and its universal covering is a triangular hyperbolic building. We can draw the link, which is a generalized 4gon, for any of these buildings: for every triple the points and , as well as and and and are incident in it. The fundamental group of the polyhedron acts simply transitively on vertices of the building. The group has 15 generators and relations, which come naturally from the polygonal presentation .
To distinguish groups , it is sufficient to distinguish the isometry classes of polyhedra, according to the Mostowtype rigidity for hyperbolic buildings which was shown, for example, in [23].
Therefore, we consider dual graphs of index 3 subgroups in order to see which of these presentations give rise to isometric polyhedra. First we calculate the index 3 subgroups: we substitute each triple of the form in the presentation by , each by three triplets , and , and each similarly by three triplets , and . We then have 45 triangles, which represent the generators of the index 3 subgroup of .
We next construct the dual graph for each of these as follows: we take 90 vertices such that first 45 of them (numbered 145) correspond to the edges of the triangles and the second 45 edges (numbered 4690) correspond to the faces of the triangles. We add an edge between vertices (from 145) and (from 4690), if edge was on the boundary of the face in a triangle. Thus we obtain trivalent bipartite graphs with 90 vertices.
With the help of the Computational Algebra System Magma we compare the dual graphs of the index 3 subgroups and we find that most of them are isomorphic with some other graph: there are only 168 nonisomorphic dual graphs. Thus we have 168 triangle presentations which give rise to nonisometric polyhedra. We then compute links of order two in buildings defined by our 168 torsion groups and 23 torsion free groups from [15]. There are only two nonisomorphic links of order two and in this case, they are complete invariants of buildings.
The 168 triangle presentations are listed in Appendix 1 together with the number (1) or (2) indicating the type of building.
This completes the proof of Theorem 2.5.
4 Construction of polyhedra with mgonal faces using torsion groups
In [15] the authors described how to construct buildings with mgonal faces, for arbitrary n, starting from torsion free groups acting on triangular buildings with the smallest possible link. We modify this construction to allow torsion groups and an arbitrary generalized polygon as the link of each vertex.
Given generalized polygon we shall denote by the graph arising by calling black resp. white vertices of black resp. white vertices of .
Consider a bipartite graph with black vertices and white vertices and a subset that defines the triangles. Starting from this triangular presentation , we construct a polyhedron, whose faces are gons and whose vertices have links or .
Let be a word of length in three letters , and . Assume that , and and that does not contain proper powers of the letters , and , that is, and for all .
For each of the triples in we take three triples , , if at least two of are different, and just one , if . The triples are cyclic, so we can write them as , and . By glueing together triangles with these words on the boundary, we obtain a polyhedron with triangle faces and 3 vertices, each of them with the graph as the link of each vertex.
We construct tuples, one corresponding to each of these new triples: for triple we define an tuple, which corresponds a word with , and . We have tuples whose coordinates start with one of the triples, and then continue with letters in some order defined by the word in the letters , , and .
If we glue the gons with these words on the boundary together by their sides labelled with same letters, respecting orientation, we obtain a polyhedron with gonal faces and vertices, which all have the link or . The type of the link can be seen from the letters of the edges meeting at that vertex. Set
and
Then for vertex the group of the link is if and if . For the last vertex we have if and if .
We denote the set of tuples by . Thus we have the following
Theorem 4.1.
The above constructed subset is a polygonal presentation. It defines a polyhedron whose faces are gons and whose vertices have links or .
Acknowledgements
The authors would like to thank Donald Cartwright and Tim Steger for many helpful discussions and an earlier (unpublished) collaboration with the first author. We are grateful to Frédéric Haglund for his interest and many illuminating discussions. Some of this work was completed at the IHÉS in the Spring of 2010. The first and third authors would like to express their gratitude to IHÉS for the Institute’s hospitality.
The first author was supported in part by NSF grant DMS1101282. The second author was supported by Emil Aaltonen Foundation, and wishes to thank École Normale Supérieure for providing excellent working conditions during Spring 2011. The last author was supported by EPSRC funded project EP/F014945/1.
Appendix 1
The torsion free cases have been listed in [15], and thus for those we only denote here in Table 2 whether the link of order 2 is isomorphic to that of (case 1) or (case 2). Then in Table 3 we list the labelings of the rows of Table 1 which give rise to the triangle presentations with torsion, denoted by . After the name of the presentation in Table 3 there is (1) resp. (2) if the resulting building is isomorphic with that of resp. .
(1)  , , , , , , , , , , , 

(2)  , , , , , , , , , , 
(1)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , , 
(2)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(1)  , , , , , , , , , , , , , ,  
(2)  , , , , , , , , , , , , , 