Posted by: matheuscmss | July 11, 2012

Homology of origamis with symmetries

Jean-Christophe Yoccoz, David Zmiaikou and I have just uploaded to ArXiv our article “Homology of origamis with symmetries“. This title resembles a (shorter) version of the title of a previous article by Jean-Christophe Yoccoz and myself (on two exceptionally symmetric origamis), and this is not mere coincidence: philosophically speaking, in the present article, we start from the point of view of D. Zmiaikou’s PhD thesis on origamis and we considerably generalize the representation-theoretical discussion in the previous article with J.-C. Yoccoz to a broader class of origamis.

Actually, a large portion of this paper was previously discussed (in details) in the posts SPCS 1, SPCS 2, SPCS 3, SPCS 4 and SPCS 5 (corresponding to some lectures given by J.-C. Yoccoz around this subject). In particular, as it was announced and explained in these posts, in this paper we mainly think of an origami (square-tiled surface) M in group-theoretical terms, i.e., M is the data (G,H,g_r,g_u) where G is a finite group generated by the elements g_r and g_u, and H is a subgroup of G containing no non-trivial normal subgroup. This point of view (“popularized” in D. Zmiaikou’s PhD thesis) is particularly useful in the study of the first homology group of a symmetric origami M (i.e., an origami with non-trivial group of automorphisms \textrm{Aut}(M)\simeq N/H:=\Gamma where N is the normalizer of H in G) because the first homology group is naturally a \Gamma-module. In other words, in this setting, the representation theory of the finite group \Gamma starts to play some role.

One of our main results is an explicit formula (that I’ll not reproduce here) for the multiplicity \ell_a of any given \Gamma-irreducible representation \chi_a in the first homology group H_1(M,\mathbb{R}). Of course, this means that we “completely” understand the homology group H_1(M,\mathbb{R}) as a \Gamma-module because there is a “general consensus” among people that the representation theory of finite groups is “well known”.

A nice feature of the formula for multiplicities \ell_a is the fact that it has a mild dependence on the data (G, H, g_r, g_u): indeed, the generators g_r and g_u enter into the formula only by means of their commutator c=[g_r,g_u]=g_r g_u g_r^{-1}g_u^{-1}. That is, we can write

H_1(M,\mathbb{R})=\bigoplus_{a\in Irr_{\mathbb{R}}(\Gamma)}W_a

where W_a\simeq\chi_a^{\ell_a} is the isotypical component of \chi_a and \ell_a is explicitly know in terms of the data G, H, N and c=[g_r,g_u]. See the first 3 sections of the article or alternatively SPCS 1, SPCS 2, and SPCS 3 for more details on this.

Then, we apply this decomposition of the homology to the study of the Kontsevich-Zorich cocycle. Roughly speaking, the idea is that the Kontsevich-Zorich cocycle is essentially the action of the group of affine diffeomorphisms of the origami (see SPCS 4 and Section 4 of the article). Since the group of affine diffeomorphisms act on the automorphism group by conjugation, it follows that the Kontsevich-Zorich cocycle permutes the isotypical components W_a in the decomposition H_1(M,\mathbb{R})=\bigoplus_{a\in Irr_{\mathbb{R}}(\Gamma)}W_a. Thus, we can pass to a finite-index subgroup of the affine group, so that isotypical components W_a are preserved. In this case, by noticing that the intersection form \{.,.\} on homology is preserved by affine diffeomorphisms, one can deduce that the Kontsevich-Zorich cocycle is essentially acting via the group Sp(W_a) of linear automorphisms of W_a preserving \{.,.\}. As it is explained in the end of Section 3 of our article (see also SPCS 3), Sp(W_a) can be completely determined in terms of the type of representation \chi_a. More precisely, it is known that real irreducible representations of a finite group are real, complex or quaternionic, and this knowledge can be used to show that:

  • Sp(W_a) is a symplectic group Sp(2d_a,\mathbb{R}) if \chi_a is real;
  • Sp(W_a) is a group of unitary complex matrices U_{\mathbb{C}}(p_a,q_a) preserving an indefinite form of signature (p_a,q_a) if \chi_a is complex;
  • Sp(W_a) is a group of unitary quaternionic matrices U_{\mathbb{H}}(p_a,q_a) preserving an indefinite form of signature (p_a,q_a) if \chi_a is quaternionic.

As it was discussed a few times in this blog (see, e.g., these links here), this has immediate consequences for the Lyapunov exponents of the Kontsevich-Zorich cocycle: indeed, the fact that Sp(W_a)\simeq U_{\mathbb{C}}(p_a,q_a) or U_{\mathbb{H}}(p_a,q_a) when \chi_a is complex or quaternionic readily implies that the presence of at least |q_a-p_a| zero Lyapunov exponents of Kontsevich-Zorich cocycle.

Remark. Actually, the relationship between the Kontsevich-Zorich cocycle and the affine diffeomorphisms is a subtle point in our article: indeed, the “usual” definition of the Kontsevich-Zorich cocycle for origamis with symmetries doesn’t lead to a dynamical linear cocycle, but merely a linear cocycle up to a finite group, i.e., a sort of orbifold linear cocycle. In particular, we spend Subsections 4.2 and 4.3 of our article to “reconciliate” the Kontsevich-Zorich cocycle with affine diffeomorphisms by taking adequate finite covers of the SL(2,\mathbb{R})-orbit of the origami.

After that, in the last section of the paper (see also SPCS 5) we use our formulas to study concrete families of origamis mainly steaming from symmetric groups G=S_n and finite groups of Lie type G=SL(2,\mathbb{F}_p).

Of course, there is no point in using this post to give further details about this paper (since this was already done in the series of posts SPCS 1 to 5). So, let me close today’s discussion by saying that we plan to address (elsewhere) the following two issues left open in our article:

  • in Subsection 5.3 of the paper, we detect the presence of complex and/or quaternionic isotypical components W_a related to regular origamis associated to the group G=SL(2,\mathbb{F}_p), but we stop our considerations right before determining the signature (p_a,q_a) of the form preserved by the restriction of Kontsevich-Zorich cocycle to W_a (of course, the motivation here comes from the study of Lyapunov exponents), and
  • even though the multiplicities \ell_a of \Gamma-irreducible representations \chi_a depend “only” on the commutator g=[g_r, g_u] (see the 4th paragraph above), we’re aware of examples of regular origamis M=(G,g_r,g_u) and M'=(G,g_r',g_u') such that c=[g_r,g_u]=[g_r',g_u'], and, a fortiori, the decomposition of the homology into W_a's is the same, but the Lyapunov exponents of M and M' are distinct: in other words, while the commutator determines the decomposition of the homology, it doesn’t determine the Lyapunov spectrum (and it is certainly interesting to investigate why this is so).
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