Hyperbolic Tessellation and Colorings of Trees

and Applied Analysis 3 0 1 1 4 1 3 2 5 1 2 3 5 2 3 3 4 1 1 6 5 4 3 3 2 5 3 9 5 2 1 9 4 12 5 5 2 8 3 11 4 3 1 (a) z 󳨃→ 3z 0 1 1 6 1 4 1 3 1


Introduction
Let  be a locally finite tree,  its vertex set, and  the set of oriented edges of .Let A be a countable set which will be called the alphabet.Let  be a coloring of , that is, a map  :  → A. Let Aut() be the automorphism group of .A periodic coloring is a coloring which is Γ-invariant for some cocompact subgroup Γ ⊂ Aut().
In this paper, we study colorings of regular trees induced from some tessellations of the hyperbolic plane.
There is a well-known family of sequences coming from rotations of circle as follows.Consider the tiling of the real line R by unit length intervals {[,  + 1) :  ∈ Z} and a map   →  +  from R to itself.There exists an integer  such that each interval [, +1) is partitioned into  or +1 subintervals of the form {[+, (+1)+) :  ∈ Z}∩{[, +1) :  ∈ Z}.Consider the sequence (  ) ∈N with   ∈ A = {, +1}, which is given by the number of such subintervals of [,  + 1).It is well known that this two-sided sequence   is periodic if only if  is rational [1].
As a generalization, we associate a coloring   of a regular tree ( ≥ 3) for any isometry  of the hyperbolic plane, given a specific hyperbolic tessellation D generated by a discrete subgroup Γ of the group of isometries on the hyperbolic plane H 2 .Suppose that each vertex of elements of D lies on the boundary of the hyperbolic plane so that the dual graph of D is a tree.
For such a tessellation D, we show that the coloring   is periodic if and only if  is a commensurator of Γ in Isom(H 2 ).
Recall that an element  ∈ Isom(H 2 ) is called a commensurator of Γ if and only if Γ −1 ∩Γ is a subgroup of Γ and of Γ −1 of finite index.Let us denote the group of commensurators of Γ by Comm(Γ).Commensurator subgroup Comm(Γ) plays an important role in the study of rigidity of locally symmetric spaces and more generally in geometric group theory ( [2][3][4]).This is a result analogous to the rotation case in the sense that the group of commensurators of SL 2 (Z) is a group containing SL 2 (Q) with finite index [5].
After showing the main theorem (Theorem 3), we show that our construction is an analogue of sequences induced from a rotation of circle only when the multiplicative constant  of   →  +  is rational.
We show that, in the case of an isometry of H 2 which is not a commensurator, we obtain colorings of unbounded alphabet, in contrast with the motivating example where irrational rotations correspond to Sturmian sequences, which are in particular sequences with a finite alphabet (see Section 3 for details).
We then explain in a heuristic way how to obtain eventually periodic colorings and colorings of "low complexity" by disregarding some information of the induced colorings.

Periodic Tree Colorings from Hyperbolic Tessellations
We first reformulate the classical example of two-sided sequences mentioned in Section 1.Consider the tessellation D of the hyperbolic plane (upper-half plane) H 2 given by the group Γ  generated by the reflections about the lines  = 0 and  = 1.More precisely, elements of D are of the form { ∈ C :  ≤ Re() <  + 1}.Then Γ  = ⟨  → −,   → 2−⟩ is isomorphic to the infinite dihedral group, and its dual graph  is a 2-regular tree.Let  = (     ) ∈ PSL 2 (R), which sends  ∈ H 2 to ( + )/( + ).Then it is not difficult to check that  is a commensurator of Γ  if and only if  = 0, and / is rational.
Let us generalize the above construction.Let us fix an ideal polygon  in the hyperbolic plane H 2 .Consider the group Γ  generated by the reflections in the edges of , which is a discrete subgroup of finite covolume in the isometry group of H 2 .By Poincaré's theorem on fundamental polygons, there exists a tessellation D of H 2 by the images of  by the elements of Γ  .
Let  be the dual graph of the tessellation D, which is a tree since  is an ideal polygon.The tree  is the Cayley graph of the group Γ  .
More generally, we will also consider the case when Γ  is generated by the reflections in the edges of a generalized ideal polygon, by which we mean a polygon in H 2 ∪ H 2 such that all vertices are on the boundary H 2 .Note that such a generalized ideal polygon may have infinite volume.
For any given  ∈ Isom(H 2 ), we associate a coloring to  as follows.Consider D = { :  ∈ D}.For any vertex  of the dual graph , the polygon   dual to  is a union of subsets of the form   ∩   , for   ∈ D, with mutually disjoint interiors.We call by the partition of  by D ∨ D the collection { ∩   :   ∈ D} just described (disregarding intersections on the boundary).

Definition 1.
Let A be the set of equivalence classes of partitions of elements of D, where two partitions of  and   , respectively, are equivalent if there exists an isometry from  to   which sends elements of the partition of  bijectively to elements of the partition of   .The coloring   associated with  is the map   :  → A sending  to the class of the partition of   by D ∨ D.
Let Γ D ⊂ Isom(H 2 ) be the set of isometries leaving D invariant.Since every  ∈ D is a generalized ideal polygon with finitely many sides, Proof.Suppose that   =   for some  ∈ Γ D ∩Γ D  −1 .Let the partition of   by D ∨ D be where {  } ∈ ⊂ D. Then Since  ∈ Γ D ∩Γ D  −1 , we have  =    −1 for some   ∈ Γ D .Thus for some {  } ∈ , {  } ∈ ⊂ D. Since   and   have the same coloring, the above partition is equal to Therefore  is a commensurator of Γ D , thus a commensurator of Γ  .Note that the coloring  #  on a 2-regular tree in (1) is periodic if   is periodic.Now let us provide some examples of isometries of the hyperbolic plane giving periodic colorings.A periodic coloring which is Γ-invariant for some Γ ⊂ Isom(H 2 ) will be expressed on the quotient, denoted by Γ \\, which is either a graph (if there is no torsion element) or a graph of groups (if there are some torsion elements, we attach the stabilizers of vertices and edges on the quotient graph).In fact, we will express a coloring on the edge-indexed graph of the quotient graph of groups Γ\\, as we only need the edge-indexed graph of a graph of groups to recover  from a graph of groups.
Recall that the edge-indexed graph of a graph of groups is a graph with an index on each oriented edge, where the graph is given by the quotient graph Γ \  and the index () of the oriented edge  is given by the index of the edge group   in the vertex group   0 () of the initial vertex  0 () of .(For details on graph of groups and the edge-indexed graph of a graph of groups, see [6][7][8].)Example 4. Consider the Farey tessellation D of the hyperbolic plane, which is the tessellation with  the ideal triangle of vertices ∞, 0, and 1.Then Γ  = ⟨−, /(2−1), 2−⟩, and the dual graph of Γ  is a 3-regular tree .Note that A hyperbolic element  1 :   → 3 and a parabolic element  2 :   →  + (1/2) is considered in Figure 1.The associated colorings   1 and   2 are both periodic.
The periodic coloring of the edge-indexed graph of a graph of groups Γ \  for   → 3 and   →  + (1/2) is as follows: 1 :  3, Here, vertices  and  represent the ideal triangle partitioned as respectively.On the other hand, In this graph, vertices  and  represent the ideal triangle partitioned as respectively (see Figure 1).Remark also that an elliptic element   → (2 − 1)/( + 1) has a periodic coloring identical to that of   → 3.

Eventually Periodic Colorings and Their Generalizations
Now consider an element of Isom(H 2 ) which is not a commensurator of Γ  .We know that the associated coloring   is not periodic.This phenomenon is in contrast to the motivating example of circle rotation explained in the beginning of the last section.In that case, irrational rotations correspond to nonperiodic colorings.However they are defined on a finite set of alphabets, and the corresponding sequences are Sturmian, that is, sequences with subword complexity () =  + 1. See [9] for Sturmian sequences.Now let us explain how to obtain colorings of "low complexity" with a finite set of alphabets from hyperbolic tessellations by disregarding some information, as the coloring on a 2-regular tree  #  in (1) for noncommensurable  is Sturmian.Definition 6.One calls a coloring  eventually periodic if there exists a subtree  of finite number of vertices such that − = ∪  is a finite union of subtrees   such that  on each   can be extended to a periodic coloring on .
In the next examples, let us denote a geodesic in H 2 between points ,  ∈ H 2 by (, ), and let us call the edges in R = H 2 boundary edges.
Example 7. Let D be the tessellation of H 2 with  a generalized ideal polygon whose edges are two geodesics (−2, 2), (−1, 1) and two boundary edges [−2, −1], [1,2].Let   = (−2  , 2  ).An element of D is a generalized ideal polygon which is the region bounded by   ,  +1 , for some  ∈ Z, which we denote by   .The dual graph is a 2-regular tree , and we can naturally denote the element of  dual to   by  ∈ Z.
In this case, the commensurator of Γ  is of the form   →  and   → / for  R. If  is considered as a map on Let  ∉ Comm(Γ  ) and  0  be a coloring given by If the boundary edge [2  , 2 +1 ] or [−2 +1 , −2  ] contains (0) (or (∞)) in its interior, then   contains   ∈  D , for sufficiently small (or large, resp.).This is the case when all vertices of   are contained in one boundary edge of   .
Otherwise, we claim that there is no  ∈ D such that  ⊂   .Indeed, suppose   is contained in   .The only remaining case is when the two boundary edges of   are contained in both of the boundary edges of   .Let  be the element of Γ sending   to   .Since  is an isometry of H 2 sending   into itself and the boundary edges of (  ) are contained in both of the boundary edges of   , it sends the geodesic segment ℓ of minimal distance between   ,  +1 , which is the intersection of the -axis with   , to a geodesic segment of minimal distance between   ,  +1 .Thus the distance between   ,  +1 is bounded above by the length of ℓ ∩   , which is strictly less than the length of ℓ, which is the distance between   and  +1 .This contradicts the fact that  is an isometry.
Therefore, all vertices except for one or two are colored by , and the remaining one or two vertices whose dual generalized ideal polygon contains (0) or (∞) in its interior are colored by .Hence, by omitting one or two vertices, one obtains a periodic coloring.Thus,  0  is an eventually periodic coloring.
In Figure 2, an example of  :   → ( − √ 11)/( √ 10( + 1)) is presented.In this case, there are exactly two vertices colored by , that is,  0  () =  for  = −2, 0 and  0  () =  otherwise.Now consider the Farey tessellation D and the corresponding group Γ.The dual graph of Γ is a 3-regular tree .Let us provide two examples of colorings given by noncommensurable elements of Γ in Isom(H 2 ).
Example 8. Let  :   →  with irrational .Then  ∉ Comm(Γ  ).Let  1  be a coloring given by for  ∈  and   ∈ D corresponding to .A geodesic line is contained in   ∩  if only if the two ideal triangles   and  have two common vertices.Since the only possible rational vertices of  are 0 and ∞,  1  () =  if and only if   corresponds to ideal triangle of vertices (0, 1, ∞) or (−1, 0, ∞).Therefore,  1   is an eventually periodic coloring, and the coloring of the edge-indexed graph of a graph of groups is as follows: For example, Figure 3 shows the case  :   → √ 3.For example, Figure 4 shows the case  :   →  + √ 17.We remark that this last example has the number of colored balls up to isometry equal to  + 2. We believe that the colorings of this type (i.e., with the number of isometry classes of colored balls being +2) are the ones corresponding to Sturmian sequences.We leave systematic studies about them for future research.
Remark 10.We can generalize the construction in this paper from torsion-free discrete subgroup to any discrete subgroup with one cusp: in this generality, one should consider the minimal subtree containing vertices not in H 2 , which is again a tree.

Figure 1 :
Figure 1: Examples of isometry  associated with periodic colorings.
Since     are all elements of D, the above partition is a partition of   by D ∨ D.Therefore, the colorings   on   and   =   are the same.Conversely, any isometry from   to   extends to an isometry of H 2 leaving D invariant.Thus if   () =   (), then there exists  ∈ Γ D such that   =   which sends elements of the partition of   by D ∨D bijectively to those of   .Let us denote the partitions of   and   =   by D ∨ D by if and only if its associated coloring   is periodic.Proof.As we mentioned earlier,  ∈ Comm(Γ  ) if and only if ∈ Comm(Γ D ).Suppose that Γ  = Γ D  −1 ∩ Γ D is a finiteindexed subgroup of Γ D .By Lemma 2, we know that   is Γ  -invariant.Since Γ D is cocompact in Aut(), Γ  is also a cocompact discrete subgroup of Aut().Thus   is periodic.Conversely, suppose that   is periodic.Let Γ be a cocompact subgroup of Aut() preserving   .For any  ∈  and  ∈ Γ we have   () =   (); thus, by Lemma 2, there exists Thus  = , and by rearranging   if necessary, we have   ∩   =   ∩   for each .Since D is a tessellation by ideal polygons, this implies that   =   .As   and   are elements of D, there exists   ∈ Γ D such that   =     .Thus   () −1 stabilizes   ; thus it is an element of Γ D , say   .We conclude that    =    −1 ∈ Γ D  −1 satisfies the statement of the lemma.Now let us formulate our theorem.Theorem 3. Let Γ  be a group generated by the reflections in the edges of a generalized ideal polygon.An isometry  ∈ Isom(H 2 ) is a commensurator of Γ

Corollary 5 .
Let   be a coloring associated with an element  ∈ Isom(H 2 ) which is not a commensurator of Γ  .Then its associated coloring has infinite alphabet.
Proof.Let Γ  = Γ D ∩ Γ D  −1 and Γ = Γ  ∩ Γ  .By Lemma 2,   () =   () implies that ,  are in the same right coset of Γ. Therefore the coloring   has a finite alphabet if and only if Γ is a finite index subgroup of Γ  .By Theorem 3, finiteness of the coloring alphabet is equivalent to the fact that  is a commensurator of Γ  .
Example 9. Let  :   →  +  with irrational .Then we have  ∉ Comm(Γ  ).Let  2  be a coloring given by , if there is  ∈ D such that   ∩  is not compact, , otherwise (14) for  ∈  and   ∈ D corresponding to .If   ∩  is not compact, then   and  have at least one common vertex.Since all vertices of D other than ∞ are irrational,  2  () =  if and only if   has the vertex of ∞, which is the only possible common vertex of   with  ∈ D.Therefore,  2 is a coloring with two colors whose coloring of the edge-indexed graph of a graph of groups is as follows: