An Upper bound on the growth of Dirichlet tilings of hyperbolic spaces
aa r X i v : . [ m a t h . G R ] J un AN UPPER BOUND ON THE GROWTH OF DIRICHLET TILINGSOF HYPERBOLIC SPACES
ITAI BENJAMINI AND TSACHIK GELANDER
Abstract.
It is shown that the growth rate (lim r | B ( r ) | /r ) of any k faces Dirichlettiling of H d , d > , is at most k − − ǫ , for an ǫ >
0, depending only on k and d .We don’t know if there is a universal ǫ u >
0, such that k − − ǫ u upperbounds thegrowth rate for any k -regular tiling, when d > Introduction
Let H d be the d -dimensional Lobachevsky space and let G = Isom( H d ) be its groupof isometries. Let Γ ≤ G be a lattice and consider an associated Dirichlet polyhedron P . Let T be the tiling of H d by Γ translates of P and let G be the dual graph of T .That is, the vertices correspond to the tiles and two vertices are connected by an edgeif they share a d − gr ( G ) = lim r | B ( r ) | /r , where | B ( r ) | is the number of vertices in an r -ball of G . The limit exists since it is asubmultiplicative sequence.Suppose that P has k faces, then the graph G is k -regular. Theorem 1.1.
For any d > and k ∈ N , there is a constant ǫ > , such that gr ( G ) < k − − ǫ for any k -regular graph dual to a finite volume convex tiling of H d . Note that the k -regular tree, T k , with k ≥ gr ( T k ) = k −
1, can berealized as a lattice graph in H . Question 1.2.
Is there a universal ǫ u > , so that any k -regular Dirichlet tiling G of H d , d > , gr ( G ) < k − − ǫ u ? There are only finitely many combinatorial types of polyhedra in dimension d with k faces, howevereach could a priory appear in infinitely many different tiling and the corresponding groups might benon-isomorphic. If such ǫ u exists, possibly k − − ǫ u will upperbound the growth rate of all tilings ofhigh dimensional simply connected homogenous spaces, which are not quasi H . Alsoassuming it exists, then maybe the growth rate k − − ǫ u is attended for some tilingwith a small k in H ?It is of interest to bound or calculate ǫ in the theorem. The argument below can beadapted to give some bounds. Remark:
The set S = { γ ∈ Γ : γ · P is adjacent to P} generates Γ and G is thecorresponding Cayley graph G = Cay(Γ , S ).Let us note that in [1] it is shown that for any ǫ > amenable Cayley graph with gr ( G ) bigger than 3 − ǫ .2. Proof
Dimension d = 3 . Here the idea is to show that the corresponding presentationof the group will have some bounded relation, of length depending on k only. Thisimplies that the growth rate is uniformly smaller than for the free group.Saying that there is a lattice whose fundamental domain is a polytope with k neigh-bours is the same as saying that the polytope has k faces and we have to specify howthe faces are glued together. This means that we have only a bounded number of ways f ( k ) of gluing and assuming it is torsion free , then it follows that there is a bound onthe number of domains glued around an edge and the wanted bound follows.For the case of torsion, if we had an edge which has no stabilizer then around thisedge there is a bound on the number of domains touching it and hence a bound on arelation length and we are done. So otherwise all edges have stabilizers which are justcyclic groups. But then consider a vertex and two edges meeting at it. We distinguishbetween the case where we could pick the vertex in the interior of the hyperbolic spaceand the case where all vertices are at infinity.In the first case, we get in SO (3) two cyclic subgroups which generate a discretegroup. These groups are of bounded size as a finite subgroup of SO (3) is either cyclic,dihedral or of order at most 60, (see e.g. [5]). In our case the group cannot be dihedralsince then one edge stabilizer would be of order 2. If it is indeed so then again we aredone.Thus we are left with the case of non-compact three dimensional lattices, wherethe fundamental domain has a vertex at infinity. Here the corresponding group willbe inside a parabolic subgroup of G , hence solvable and by discreteness will preserve N UPPER BOUND ON THE GROWTH OF DIRICHLET TILINGS OF HYPERBOLIC SPACES 3 horospheres and hence must be one of the finitely many non-cyclic discrete subgroupsIsom( R ), and the argument continues like before.2.2. Dimension d ≥ . Consider now the case of dimension strictly bigger than three.By induction on k one shows that there is a triangulation of bounded size (easilycomputable) of the Dirichlet fundamental domain. Since hyperbolic simplices admitan upper bound on their volume, this gives a bound on the volume. Now since d > γ in a bounded power l of S which stabilizes it.Suppose first that G/ Γ is not compact. In that case P has a vertex at infinity ζ .Let γ , γ ∈ S l be bounded elements stabilizing two distinct co-dim 2 walls through ζ .Then γ , γ belongs to a spherical crystallographic subgroup of Γ, hence by Bieberbachtheorem (see e.g. [6]) satisfies [ γ m , γ m ] = 1 where m depends only on d −
1. Thus,there is a relation of bounded length, which implies the growth gap in this case.Next suppose that G/ Γ is compact. Here again one can argue similarly to the 3-dimensional case, however we give a different argument which can clearly be carriedout in a much more general situation (when Wang’s finiteness theorem applies). Notethat a Dirichlet domain is determined by a point, and up to equivalence, it is enoughto pick that point in a given fundamental domain, say P ′ .Let D = diam( P ′ ) and set ˜ P ′ = N D ( P ′ ), the closed D -neighborhood of P ′ . Then ˜ P ′ is compact, hence the set ˜ S := { γ ∈ Γ : γ · ˜ P ′ ∩ ˜ P ′ = ∅} is finite.Any Γ-periodic k -regular graph G associated with a uniform Dirichlet tiling of H d is isomorphic to the Cayley graph Cay(Γ : S ) for some S ⊂ ˜ S , of cardinality k . Thereare only finitely many such graphs, and these graphs are all transitive and not a tree.The result follows, since | B ( r ) | is a submultiplicative sequence. (cid:3) Further problems and remarks (1) Let L ( k, d ) be the set of k -regular dual graphs for Dirichlet tilings in H d and gr ( k, d ) the set of growth rate of these graphs. Are there non-isomorphic such ITAI BENJAMINI AND TSACHIK GELANDER graphs with the same growth rate. Bound the number of elements in thesesets. Are the values in gr ( k, d ) rational or algebraic? How big/small can theexponents in gr ( k, d ) be?(2) We conjecture that the maximal possible growth rate of k -regular dual graphof a Dirichlet tiling, is monotone decreasing in the dimension (as long as k > d ,for larger k there is no such graph).(3) A similar result should hold considering lattices in general real symmetricspaces of non-compact type of dimension greater than 2, in particular theargument for compact Dirichlet domain that we gave for d ≥ L betti number, or property T .What is the natural general statement?(5) In H d , d >
3, are there only finitely many non-isomorphic such k -regular graphsfor any k , estimate this number?(6) We studied here only Dirichlet tilings. We believe that similar bounds shouldhold for more general tilings. Including Voronoi tilings of point process suchas Poisson point process and aperiodic tilings. See [2, 3, 4] for constructionsof aperiodic tiling of the hyperbolic plane. There is no construction yet inhyperbolic spaces of higher dimension and other symmetric spaces, such as H × R or H × H .(7) Given the tiling graph, from every vertex pick a geodesic to the root. Do it byfirst connecting ball of radius 1 then from all vertices at distance 2 to verticesat distance 1, and so on. Look at the spanning subgraph consists of all verticesand only edges that are on such a chosen geodesic. This graph has the samevolume growth as the tiling graph. Still maybe one can show directly thatthere is a positive portion of the edges that are not on this geodesic tree, atabout every distance. A direct argument might work beyond Dirichlet tilings. Acknowledgements: we are grateful to Anton Malyshev and Shahar Mozes.
References [1] G. Arzhantseva, V. Guba and L. Guyot, Growth rates of amenable groups. J. Group Theory 8(2005) 389-394.
N UPPER BOUND ON THE GROWTH OF DIRICHLET TILINGS OF HYPERBOLIC SPACES 5 [2] C. Goodman-Strauss, A hierarchical strongly aperiodic set of tiles in the hyperbolic plane.Theoret. Comput. Sci. 411 (2010) 1085-1093.[3] C. Goodman-Strauss, A strongly aperiodic set of tiles in the hyperbolic plane. Invent. Math.159 (2005) 119-132.[4] G. Margulis and S. Mozes, Aperiodic tilings of the hyperbolic plane by convex polygons. IsraelJ. Math. 107 (1998) 319-325.[5] H. Mark, Classifying finite subgroups of SO ∼∼