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) in group-theoretical terms, i.e., is the data where is a finite group generated by the elements and , and is a subgroup of 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 (i.e., an origami with non-trivial group of automorphisms where is the normalizer of in ) because the first homology group is naturally a -module. In other words, in this setting, the representation theory of the finite group starts to play some role.
One of our main results is an explicit formula (that I’ll not reproduce here) for the multiplicity of any given -irreducible representation in the first homology group . Of course, this means that we “completely” understand the homology group as a -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 is the fact that it has a mild dependence on the data : indeed, the generators and enter into the formula only by means of their commutator . That is, we can write
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 in the decomposition . Thus, we can pass to a finite-index subgroup of the affine group, so that isotypical components 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 of linear automorphisms of preserving . As it is explained in the end of Section 3 of our article (see also SPCS 3), can be completely determined in terms of the type of representation . 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:
- is a symplectic group if is real;
- is a group of unitary complex matrices preserving an indefinite form of signature if is complex;
- is a group of unitary quaternionic matrices preserving an indefinite form of signature if 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 or when is complex or quaternionic readily implies that the presence of at least 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 -orbit of the origami.
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 related to regular origamis associated to the group , but we stop our considerations right before determining the signature of the form preserved by the restriction of Kontsevich-Zorich cocycle to (of course, the motivation here comes from the study of Lyapunov exponents), and
- even though the multiplicities of -irreducible representations depend “only” on the commutator (see the 4th paragraph above), we’re aware of examples of regular origamis and such that , and, a fortiori, the decomposition of the homology into s is the same, but the Lyapunov exponents of and 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).