Posted by: matheuscmss | January 18, 2012

“Surfaces à petits carreaux (suite)” by Jean-Christophe Yoccoz

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.

-Introduction-

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 \pi:M\to\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2 where M is a connected surface and \pi is not ramified on \mathbb{T}^2-\{0\}.

The squares are the connected components of \pi^{-1}((0,1)^2). We denote by Sq(M) the set of squares.

A morphism p between two origamis \pi:M\to\mathbb{T}^2 and \pi':M'\to\mathbb{T}^2 is an application p:M\to M' such that \pi = \pi'\circ p. An isomorphism is a homeomorphism p such that \pi = \pi'\circ p.

2. Geometrical-Analytical point of view

We begin by reviewing the notion of translation structures. Let M be a compact connected orientable surface genus g\geq 1, and \emptyset\neq\Sigma=\{A_1,\dots,A_s\}\subset M, and \kappa=(k_1,\dots,k_s)\in\mathbb{N}^s, k_i\geq 1, \sum\limits_{i=1}^s k_i=2g-2.

A structure of translation surface on (M,\Sigma) of type \kappa is a maximal atlas \zeta of local charts on M-\Sigma verifying:

  • the coordinate changes are locally given by translations;
  • for each 1\leq i\leq s, there are neighborhoods V_i of A_i\in M, W_i of 0\in\mathbb{R}^2 and a ramified covering \pi_i:(V_i,A_i)\to(W_i,0) of degree k_i such that the injective restrictions of \pi_i are local charts of \zeta (in particular, the “total angle” around A_i is 2\pi k_i).

Equivalently, we can say that a translation surface is a Riemann surface structure on M together with a choice of a non-trivial holomorphic 1-form \omega\neq 0 with a zero of order (k_i-1) at A_i. Indeed, this last definition can be connected to the previous one by noticing that the local primitives of \omega provide local charts for an atlas \zeta as above.

Next, we recall that, given a translation surface structure, one has a period map \Theta:H_1(M,\Sigma,\mathbb{Z})\to\mathbb{R}^2=\mathbb{C}, \Theta([\gamma]):=\int_{\gamma}\omega.

In this setting, an origami is a translation surface whose period map \Theta takes values in \mathbb{Z}^2.

Remark (ambiguity on \Sigma). In the previous definition, the set \Sigma was not explicitly chosen. In fact, by letting \Sigma_{\min}=\{\textrm{ramification points of }\pi:M\to\mathbb{T}^2\} and \Sigma_{\max}:=\pi^{-1}(\{0\}), we will see that any choice of \Sigma_{\min}\subset\Sigma\subset\Sigma_{\max} works.

In order to see that the geometrical-analytical definition of origami is equivalent to the topological one, we take A_1\in\Sigma, and we observe that \pi(z):=\int_{A_1}^z\omega (mod \mathbb{Z}^2) is a well-defined topological origami and, conversely, a topological origami \pi:M\to\mathbb{T}^2 defines a geometrical-analytical origami by taking the inverses of injective restrictions of \pi as the local charts of an atlas \zeta with the desired properties.

3. Algebraic-Combinatorial point of view

An origami is a finite set \mathcal{O} equipped with two permutations r, u\in S(\mathcal{O}) generating a transitive action on \mathcal{O}. Given a topological origami \pi:M\to\mathbb{T}^2, we set \mathcal{O}=Sq(M) and r(s):=\textrm{square to the right of }s, u(s):=\textrm{square on the top of }s. In this way, the connectedness of M corresponds to the transitivity of the action of r,u on \mathcal{O}.

Conversely, given an algebraic-combinatorial origami, we see that M=(\mathcal{O}\times [0,1]^2)/\sim, where \sim is the equivalence relation (s,1,x)\sim (r(s),0,x) and (s,x,1)\sim (u(s),x,0), is a topological origami.

A morphism between (\mathcal{O},r,u) and (\mathcal{O}',r',u') is an application p:\mathcal{O}\to\mathcal{O}' such that p\circ r = r'\circ p and p\circ u = u'\circ p. An isomorphism is a bijective morphism p.

-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 Mon(\mathcal{O}) is the subgroup of the permutation group S(\mathcal{O}) generated by r and u.

Convention. We will let S(\mathcal{O}) acts on the right on \mathcal{O}, i.e., (sg)g' = s(gg') (because we want automorphisms of origamis to act on the left).

1. Ramification

The type \kappa is determined by the action of the commutator c=r^{-1}u^{-1}ru:

In fact, from the figure above one deduces that s = s\cdot c^{k(s)} where k(s) is the ramification index of the left-down corner of the square s. (Warning: the figure above is somewhat misleading as s and s\cdot c are usually not the same square!).

2. Origamis and quotients of groups

Denote by G=Mon(\mathcal{O}) and let g_r, g_u\in G be the elements corresponding to r,u. Fix \star\in\mathcal{O}, and denote by H the stabilizer of \star with respect to the (right-) action of G on \mathcal{O}.

We identify \mathcal{O} = H\backslash G=\{Hg: g\in G\}, \star = H and r:Hg\mapsto Hgg_r, u: Hg\mapsto Hgg_u.

Note that H is not an arbitrary subgroup of G. Indeed, since the stabilizer of Hg is g^{-1}Hg, the transitivity faithfulness of the action implies that the intersection of the conjugates of H is the trivial group \{1\}. In other words, H doesn’t contain non-trivial normal subgroups.

Conversely, given a finite group G generated by two elements g_r,g_u\in G, and H\subset G a subgroup containing no non-trivial normal subgroup, we obtain an origami M by taking \mathcal{O}=H\backslash G and r:Hg\mapsto Hgg_r, u: Hg\mapsto Hgg_u.

Of course, in this language, a change of basepoint \star corresponds to replace H by one of its conjugates g^{-1}Hg.

3. Automorphisms of an origami

By definition, an automorphism is a bijection p:\mathcal{O}\to\mathcal{O} such that p\circ r = r\circ p and p\circ u = u\circ p.

Denote by n\in G an element with H\cdot n = p(H) (=p(\star). For every g\in G, one has Hng=p(Hg) (since p is an automorphism). Thus, for every h\in H, Hn=p(H)=Hnh, i.e., nhn^{-1}\in H, i.e., n belongs to the normalizer N of H in G. In particular, for every g\in G, one has p(Hg)=Hng=nHg, i.e., the automorphisms of M are given by the left action of the normalizer N of H in G (see the convention above). So, by putting n\cdot Hg = nHg=Hng, we can identify

Aut(M)\simeq N/H

Definition (D.Zmiaikou). We say that M is a regular origami if Aut(M) acts transitively on Sq(M), or, equivalently, H=\{1\} (and, a fortiori, N=G).

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 Aut(M) on H_1(M,K) with K=\mathbb{Q},\mathbb{R},\mathbb{C};
  • consequences of the previous topic to the action of the affine group Aff(M) on H_1(M,K) 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 M as a Aut(M)-module-

Let K be a subfield of \mathbb{C}. Given an origami \pi:M\to\mathbb{T}^2, take \Sigma=\Sigma_{\max}=\pi^{-1}(\{0\}) and consider the first relative homology group

H_1(M,\Sigma,K)

Recall that Sq(M)=H\backslash G, G=\langle g_r, g_u\rangle, and \bigcap\limits_{g\in G} g^{-1}Hg=\{1\}. We denote by K(M)=K^{H\backslash G} the K-vector space of canonical basis e_s, s\in H\backslash G. Next, we take two copies K(M)\oplus K(M) of K(M) and we denote by \sigma_s, s\in H\backslash G, the canonical basis of the first copy, and by \zeta_s, s\in H\backslash G, the canonical basis of the second copy.

Proposition. We have an exact sequence of Aut(M)-modules

0\to K\stackrel{\varepsilon}{\to} K(M)\stackrel{i}{\to} K(M)\oplus K(M)\stackrel{j}{\to} H_1(M,\Sigma,K)\to 0

where \varepsilon(1)=\sum\limits_{s\in H\backslash G} e_s, i(e_s)=\square_s:=\sigma_s + \zeta_{sg_r} - \sigma_{s g_u} - \zeta_s, and the map j is explained in the picture below:

Proof. Let Sk(M) = M-\bigcup\limits_{s\in H\backslash G}s be the skeleton of M. We have that \Sigma\subset Sk(M)\subset M gives rise to an exact sequence in homology:

H_2(Sk(M),\Sigma,K)\to H_2(M,\Sigma,K)\to H_2(M,Sk(M),K)\stackrel{\partial}{\to}

H_1(Sk(M),\Sigma,K)\to H_1(M,\Sigma,K)\to H_1(M,Sk(M),K)

where \partial is the usual boundary operator.

Since Sk(M) is a 1-dimensional skeleton, H_2(Sk(M),\Sigma,K)=0 and H_1(M,Sk(M),K)=0, H_2(M,Sk(M),K)=K(M) and H_1(Sk(M),\Sigma,K)=K(M)\oplus K(M). Also, since M is 2-dimensional, H_2(M,Sk(M),K)=K(M). 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. \square

Corollary. One has H_1(M,\Sigma,K)=K(M)\oplus K as Aut(M)-module. \square

We recall that one can write a decomposition of Aut(M)-modules:

H_1(M,K)=H_1^{st}(M,K)\oplus H_1^{(0)}(M,K)

where H_1^{st}(M,K) is the 2-dimensional Aut(M)-module generated by \sum\limits_{s\in H\backslash G}\sigma_s and \sum\limits_{s\in H\backslash G}\zeta_s (hence it is isomorphic to K^2) and H_1^{(0)}(M,K) is the codimension 2 (i.e., dimension 2g-2) given by the kernel of the map

\pi_*:H_1(M,K)\to H_1(\mathbb{T}^2,K)=K^2,

that is, H_1^{(0)}(M,K) consists of homology classes projecting to zero in the torus.

On the other hand, the exact sequence in homology associated to \emptyset\subset\Sigma\subset M is

0=H_1(\Sigma,K)\to H_1(M,K)\to H_1(M,\Sigma,K)\to

H_0(\Sigma,K)\to H_0(M,K)=K\to H_0(\emptyset,K)=0

By combining this with the previous corollary, we get

Corollary. H_1^{(0)}(M,K)\simeq K(M)\ominus H_0(\Sigma,K) as Aut(M)-module. \square

As we told above, our current goal is to understand the action of Aut(M) on H_1(M,K). Of course, since the action of Aut(M) on H_1^{st}(M,K) is trivial, it suffices to understand the action of Aut(M) on H_1^{(0)}(M,K), and, by the previous corollary, this amounts to study H_0(\Sigma,K)=K^{\Sigma}. Therefore, we’ll close today’s post with some preliminaries on the action of Aut(M)=N/H on \Sigma.

Denote by k the order of the commutator c=g_r^{-1}g_u^{-1}g_r g_u, and let \langle c\rangle be the subgroup generated by it. During the study of ramifications of origamis in terms of the commutator, we saw that the points of \Sigma correspond to orbits of the action of \langle c\rangle on H\backslash G, or, equivalently, orbits of the action of H\times\langle c\rangle on G given by (h,c^m)\cdot g\mapsto hgc^m.

In this way, for each g\in G, we denote by A_g the point of \Sigma corresponding to the action of g, and Stab(g) the stabilizer of A_g for the action of N on \Sigma.

Lemma. Stab(g)=N\cap H\cdot\langle gcg^{-1}\rangle = H\cdot (N\cap\langle gcg^{-1}\rangle).

Proof. Observe that

n\in Stab(g)\iff nHg\langle c\rangle=Hg\langle c\rangle\iff Hng\langle c\rangle=Hg\langle c\rangle\iff

n\in H\langle gcg^{-1}\rangle\iff n=h \underbrace{gc^mg^{-1}}_{\substack{\in N \\ \textrm{ since } n,h\in N}}\iff n\in H\cdot (N\cap\langle gcg^{-1}\rangle)

This completes the proof. \square

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 Aut(M) on H_1^{(0)}(M,K).


Responses

  1. Hi Matheus,

    An error in section 2 ? The fact that H does not contain a normal subgroup does not follow from “transitivity” of the action but from “faithfulness”. More precisely, the intersection of g^-1 H g for g in G is exactly the set of elements in G that acts as the identity on H \ G. It is also the maximal normal subgroup of G contained in H.

    In section 3. you did not insist on the fact that the automorphism group is defined as a centralizer. Actually, you prove a nice group theoretical statement : the centralizer in Sym(H \ G) of G is isomorphic to N_G(H) / H.

    V. D.

  2. Hi Vincent,

    Thanks for the comments! In fact, you’re right that it is faithfulness that we’re using in section 2, and that the argument showing that the automorphism group is N/H in section 3 translates into the fact that the centralizer of G in Aut(M) is N/H (but I confess that I did not insist on this because it was part of the material of last year’s course)

    Best,

    Matheus


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