From January 11 to March 21, Jean-Christophe Yoccoz delivers (on Wednesdays) his course (of academic year 2011-2012) at Collège de France. As the reader can find in his webpage, he decided to make a continuation of his last course (about square-tiled surfaces) and so he entitled the current series of lectures “Surfaces à petits carreaux (suite)”.
After following the first two lectures, I thought it could be a nice idea to try to make available the notes I’m taking for this course. So, I plan to write a series of posts whose titles will have the form “SPCS x” (where SPCS stands for“Surfaces à petits carreaux (suite)” and “x” stands for the number of the lecture). Of course, this goes without saying that any errors and/or mistakes are surely my sole fault: indeed, since the course is delivered in French, it may happen that I misinterpret some points.
Below the fold the reader will find my first set of notes, i.e., SPCS 1, corresponding to Yoccoz’s lecture on last January 11th.
We start by recalling some distinct (but completely equivalent) points of view (discussed in last year’s course) on square-tiled surfaces / origamis.
1. Topological point of view
An origami is a finite (usually ramified) covering where is a connected surface and is not ramified on .
The squares are the connected components of . We denote by the set of squares.
A morphism between two origamis and is an application such that . An isomorphism is a homeomorphism such that .
2. Geometrical-Analytical point of view
We begin by reviewing the notion of translation structures. Let be a compact connected orientable surface genus , and , and , , .
A structure of translation surface on of type is a maximal atlas of local charts on verifying:
- the coordinate changes are locally given by translations;
- for each , there are neighborhoods of , of and a ramified covering of degree such that the injective restrictions of are local charts of (in particular, the “total angle” around is ).
Equivalently, we can say that a translation surface is a Riemann surface structure on together with a choice of a non-trivial holomorphic -form with a zero of order at . Indeed, this last definition can be connected to the previous one by noticing that the local primitives of provide local charts for an atlas as above.
Next, we recall that, given a translation surface structure, one has a period map , .
In this setting, an origami is a translation surface whose period map takes values in .
Remark (ambiguity on ). In the previous definition, the set was not explicitly chosen. In fact, by letting and , we will see that any choice of works.
In order to see that the geometrical-analytical definition of origami is equivalent to the topological one, we take , and we observe that (mod ) is a well-defined topological origami and, conversely, a topological origami defines a geometrical-analytical origami by taking the inverses of injective restrictions of as the local charts of an atlas with the desired properties.
3. Algebraic-Combinatorial point of view
An origami is a finite set equipped with two permutations generating a transitive action on . Given a topological origami , we set and , . In this way, the connectedness of corresponds to the transitivity of the action of on .
Conversely, given an algebraic-combinatorial origami, we see that , where is the equivalence relation and , is a topological origami.
A morphism between and is an application such that and . An isomorphism is a bijective morphism .
-Monodromy of origamis and description of origamis as quotients of groups-
This algebraic-combinatorial point of view of origamis was studied by D. Zmiaikou in his PhD thesis (under the supervision of J.-C. Yoccoz). In the sequel, we will follow closely some aspects of D. Zmiaikou’s thesis.
Definition (D. Zmiaikou). The monodromy group is the subgroup of the permutation group generated by and .
Convention. We will let acts on the right on , i.e., (because we want automorphisms of origamis to act on the left).
The type is determined by the action of the commutator :
In fact, from the figure above one deduces that where is the ramification index of the left-down corner of the square . (Warning: the figure above is somewhat misleading as and are usually not the same square!).
2. Origamis and quotients of groups
Denote by and let be the elements corresponding to . Fix , and denote by the stabilizer of with respect to the (right-) action of on .
We identify , and , .
Note that is not an arbitrary subgroup of . Indeed, since the stabilizer of is , the
transitivity faithfulness of the action implies that the intersection of the conjugates of is the trivial group . In other words, doesn’t contain non-trivial normal subgroups.
Conversely, given a finite group generated by two elements , and a subgroup containing no non-trivial normal subgroup, we obtain an origami by taking and , .
Of course, in this language, a change of basepoint corresponds to replace by one of its conjugates .
3. Automorphisms of an origami
By definition, an automorphism is a bijection such that and .
Denote by an element with . For every , one has (since is an automorphism). Thus, for every , , i.e., , i.e., belongs to the normalizer of in . In particular, for every , one has , i.e., the automorphisms of are given by the left action of the normalizer of in (see the convention above). So, by putting , we can identify
Definition (D.Zmiaikou). We say that is a regular origami if acts transitively on , or, equivalently, (and, a fortiori, ).
The general plan for the beginning of the course is to explain some results from a work (still in progress) by J.-C. Yoccoz, D. Zmiaikou and myself, where the following topics are studied:
- the action of on with ;
- consequences of the previous topic to the action of the affine group on and the Lyapunov exponents of the Kontsevich-Zorich cocycle;
- application of the previous topics to concrete examples.
In particular, the results we’re going to discuss below (and in the next two or three lectures) are part of a forthcoming paper by J-C. Y., D. Z., C. M.
-Homology of a origami as a -module-
Let be a subfield of . Given an origami , take and consider the first relative homology group
Recall that , , and . We denote by the -vector space of canonical basis , . Next, we take two copies of and we denote by , , the canonical basis of the first copy, and by , , the canonical basis of the second copy.
Proposition. We have an exact sequence of -modules
where , , and the map is explained in the picture below:
Proof. Let be the skeleton of . We have that gives rise to an exact sequence in homology:
where is the usual boundary operator.
Since is a -dimensional skeleton, and , and . Also, since is -dimensional, . By plugging this information into the previous exact sequence (and by reinterpreting the operators between the homology groups above), one can check that this exact sequence is precisely the one appearing in the conclusion of the proposition.
Corollary. One has as -module.
We recall that one can write a decomposition of -modules:
where is the -dimensional -module generated by and (hence it is isomorphic to ) and is the codimension (i.e., dimension ) given by the kernel of the map
that is, consists of homology classes projecting to zero in the torus.
On the other hand, the exact sequence in homology associated to is
By combining this with the previous corollary, we get
Corollary. as -module.
As we told above, our current goal is to understand the action of on . Of course, since the action of on is trivial, it suffices to understand the action of on , and, by the previous corollary, this amounts to study . Therefore, we’ll close today’s post with some preliminaries on the action of on .
Denote by the order of the commutator , and let be the subgroup generated by it. During the study of ramifications of origamis in terms of the commutator, we saw that the points of correspond to orbits of the action of on , or, equivalently, orbits of the action of on given by .
In this way, for each , we denote by the point of corresponding to the action of , and the stabilizer of for the action of on .
Proof. Observe that
This completes the proof.
At this point, Jean-Christophe ended his lecture (as he was running out of time), and, so we also stop here for today. Next time, we will continue the study of the action of on .