Abstract
The features of the \(SL(2,\mathbb {R})\)-action on the moduli spaces of translation surfaces (and its applications to the study of interval exchange transformations, translation flows and billiards) are intimately related to the properties of the so-called Kontsevich-Zorich (KZ) cocycle. In particular, it is not surprising that the KZ cocycle is one of the main actors in the recent groundbreaking work of Eskin and Mirzakhani spsciteEsMi towards the classification of \(SL(2,\mathbb {R})\)-invariant measures on moduli spaces of translation surfaces.
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Notes
- 1.
See [26, Sect. 3.2].
- 2.
In concrete terms, \(SO^*(2n)\) is the group of \(n\times n\) matrices A with coefficients in the quaternions \(\mathbf H \) such that \(A^{\#} A = \text {Id}\) where \(A^{\#}\) is the transpose of \(\sigma (A)\) and \(\sigma (a+bi+cj+dk)=a-bi+cj+dk\) is a reversion on \(\mathbf H \).
- 3.
E.g., by Hurwitzs theorem saying that a Riemann surface of genus \(g\ge 2\) has \(84(g-1)\) automorphisms at most.
- 4.
I.e., up to finite-index, the KZ cocycle commutes with the action of Q on \(H_1(\widetilde{L},\mathbb {R})\).
- 5.
A product of three copies of the compact group \(SO^*(2)\).
- 6.
I.e., the matrix has an unstable (modulus \(>1\)) eigenvalue, a central (modulus \(=1\)) eigenvalue, and an stable (modulus \(<1\)) eigenvalue, all of them with multiplicity four.
- 7.
Actually, \(\{v_{AB}^{(n)}, v_{CB}^{(m)}\}_{1\le n, m\le 4}\) span a 8-dimensional vector space.
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Matheus Silva Santos, C. (2018). An Example of Quaternionic Kontsevich-Zorich Monodromy Group. In: Dynamical Aspects of Teichmüller Theory. Atlantis Studies in Dynamical Systems, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-92159-4_6
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DOI: https://doi.org/10.1007/978-3-319-92159-4_6
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