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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 706496, 6 pages
Hyperbolic Tessellation and Colorings of Trees
1Department of Mathematics Education, Dongguk University-Seoul, Seoul 100-715, Republic of Korea
2Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea
Received 22 February 2013; Accepted 3 May 2013
Academic Editor: Baodong Zheng
Copyright © 2013 Dong Han Kim and Seonhee Lim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study colorings of a tree induced from isometries of the hyperbolic plane given an ideal tessellation. We show that, for a given tessellation of the hyperbolic plane by ideal polygons, a coloring can be associated with any element of Isom(), and the element is a commensurator of if and only if its associated coloring is periodic, generalizing a result of Hedlund and Morse.
Let be a locally finite tree, its vertex set, and the set of oriented edges of . Let be a countable set which will be called the alphabet. Let be a coloring of , that is, a map . Let be the automorphism group of . A periodic coloring is a coloring which is -invariant for some cocompact subgroup .
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 by unit length intervals and a map from to itself. There exists an integer such that each interval is partitioned into or subintervals of the form . Consider the sequence with , which is given by the number of such subintervals of . It is well known that this two-sided sequence is periodic if only if is rational .
As a generalization, we associate a coloring of a -regular tree () for any isometry of the hyperbolic plane, given a specific hyperbolic tessellation generated by a discrete subgroup of the group of isometries on the hyperbolic plane . Suppose that each vertex of elements of lies on the boundary of the hyperbolic plane so that the dual graph of is a tree.
For such a tessellation , we show that the coloring is periodic if and only if is a commensurator of in . Recall that an element is called a commensurator of if and only if is a subgroup of and of of finite index. Let us denote the group of commensurators of by . Commensurator subgroup plays an important role in the study of rigidity of locally symmetric spaces and more generally in geometric group theory ([2–4]).
This is a result analogous to the rotation case in the sense that the group of commensurators of is a group containing with finite index .
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 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.
2. Periodic Tree Colorings from Hyperbolic Tessellations
We first reformulate the classical example of two-sided sequences mentioned in Section 1. Consider the tessellation of the hyperbolic plane (upper-half plane) given by the group generated by the reflections about the lines and . More precisely, elements of are of the form . Then is isomorphic to the infinite dihedral group, and its dual graph is a 2-regular tree. Let , which sends to . Then it is not difficult to check that is a commensurator of if and only if , and is rational.
Let us generalize the above construction. Let us fix an ideal polygon in the hyperbolic plane . Consider the group generated by the reflections in the edges of , which is a discrete subgroup of finite covolume in the isometry group of . By Poincaré’s theorem on fundamental polygons, there exists a tessellation of by the images of by the elements of .
Let be the dual graph of the tessellation , which is a tree since is an ideal polygon. The tree is the Cayley graph of the group .
More generally, we will also consider the case when is generated by the reflections in the edges of a generalized ideal polygon, by which we mean a polygon in such that all vertices are on the boundary . Note that such a generalized ideal polygon may have infinite volume.
For any given , we associate a coloring to as follows. Consider . For any vertex of the dual graph , the polygon dual to is a union of subsets of the form , for , with mutually disjoint interiors. We call by the partition of by the collection just described (disregarding intersections on the boundary).
Definition 1. Let be the set of equivalence classes of partitions of elements of , 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 sending to the class of the partition of by .
Let be the set of isometries leaving invariant. Since every is a generalized ideal polygon with finitely many sides, is a finite index subgroup of . Thus is a commensurator of if and only if is a commensurator of .
Lemma 2. For each , one has if and only if there exists such that , where , are the elements of associated with .
Proof. Suppose that for some . Let the partition of by be
where . Then
Since , we have for some . Thus
Since are all elements of , the above partition is a partition of by . Therefore, the colorings on and are the same.
Conversely, any isometry from to extends to an isometry of leaving invariant. Thus if , then there exists such that which sends elements of the partition of by bijectively to those of . Let us denote the partitions of and by by for some . Since and have the same coloring, the above partition is equal to Thus , and by rearranging if necessary, we have for each . Since is a tessellation by ideal polygons, this implies that . As and are elements of , there exists such that . Thus stabilizes ; thus it is an element of , say . We conclude that satisfies the statement of the lemma.
Now let us formulate our theorem.
Theorem 3. Let be a group generated by the reflections in the edges of a generalized ideal polygon. An isometry is a commensurator of if and only if its associated coloring is periodic.
Proof. As we mentioned earlier, if and only if . Suppose that is a finite-indexed subgroup of . By Lemma 2, we know that is -invariant. Since is cocompact in , is also a cocompact discrete subgroup of . Thus is periodic.
Conversely, suppose that is periodic. Let be a cocompact subgroup of preserving . For any and we have ; thus, by Lemma 2, there exists such that . Letting , it follows that is finite since is finite. Since is a finite index subgroup of and is the Cayley graph of , is of finite index. Therefore is a commensurator of , thus a commensurator of .
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 -invariant for some will be expressed on the quotient, denoted by , 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 , 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 and the index of the oriented edge is given by the index of the edge group in the vertex group of the initial vertex of . (For details on graph of groups and the edge-indexed graph of a graph of groups, see [6–8].)
Example 4. Consider the Farey tessellation of the hyperbolic plane, which is the tessellation with the ideal triangle of vertices , , and . Then , and the dual graph of is a 3-regular tree . Note that is a subgroup of of index 2. is commensurable to as is a subgroup of of index six since is the stabilizer subgroup of in .
A hyperbolic element and a parabolic element is considered in Figure 1. The associated colorings and are both periodic.
The periodic coloring of the edge-indexed graph of a graph of groups for and is as follows: (7) Here, vertices and represent the ideal triangle partitioned as (8) respectively. On the other hand, (9) In this graph, vertices and represent the ideal triangle partitioned as (10) respectively (see Figure 1).
Remark also that an elliptic element has a periodic coloring identical to that of .
3. Eventually Periodic Colorings and Their Generalizations
Now consider an element of which is not a commensurator of . We know that the associated coloring is not periodic.
Corollary 5. Let be a coloring associated with an element which is not a commensurator of . Then its associated coloring has infinite alphabet.
Proof. Let and . By Lemma 2, implies that , are in the same right coset of . Therefore the coloring has a finite alphabet if and only if is a finite index subgroup of . By Theorem 3, finiteness of the coloring alphabet is equivalent to the fact that is a commensurator of .
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 . See  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 between points by , and let us call the edges in boundary edges.
Example 7. Let be the tessellation of with a generalized ideal polygon whose edges are two geodesics , and two boundary edges , . Let . An element of is a generalized ideal polygon which is the region bounded by , , for some , which we denote by . The dual graph is a 2-regular tree , and we can naturally denote the element of dual to by .
In this case, the commensurator of is of the form and for . If is considered as a map on , then if and only if .
Let and be a coloring given by
If the boundary edge or contains (or ) in its interior, then contains , 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 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 sending to . Since is an isometry of sending into itself and the boundary edges of are contained in both of the boundary edges of , it sends the geodesic segment of minimal distance between , , which is the intersection of the -axis with , to a geodesic segment of minimal distance between , . Thus the distance between , is bounded above by the length of , which is strictly less than the length of , which is the distance between and . 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 or in its interior are colored by . Hence, by omitting one or two vertices, one obtains a periodic coloring. Thus, is an eventually periodic coloring.
In Figure 2, an example of is presented. In this case, there are exactly two vertices colored by , that is, for , and otherwise.
Now consider the Farey tessellation and the corresponding group . The dual graph of is a 3-regular tree . Let us provide two examples of colorings given by non-commensurable elements of in .
Example 8. Let with irrational . Then . Let be a coloring given by
for and 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 , if and only if corresponds to ideal triangle of vertices or . Therefore, is an eventually periodic coloring, and the coloring of the edge-indexed graph of a graph of groups is as follows:(13)
For example, Figure 3 shows the case .
Example 9. Let with irrational . Then we have . Let be a coloring given by
for and corresponding to . If is not compact, then and have at least one common vertex. Since all vertices of other than are irrational, if and only if has the vertex of , which is the only possible common vertex of with . Therefore, is a coloring with two colors whose coloring of the edge-indexed graph of a graph of groups is as follows:(15)
For example, Figure 4 shows the case .
We remark that this last example has the number of colored balls up to isometry equal to . We believe that the colorings of this type (i.e., with the number of isometry classes of colored balls being ) 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 , which is again a tree.
The authors would like to thank the anonymous referee for the valuable comments. The first author is supported by NRF 2012R1A1A2004473. The second author is supported by NRF 2012-000-8829, NRF 2012-000-2388, and TJ Park Science Fellowship.
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