SQUARE-TILED CYCLIC COVERS

By G. Forni, Carlos Matheus, A. Zorich. 2010

55 citations recorded by the source corpus.

A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichmuller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichmuller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in [17] of a Teichmuller curve of a square-tiled cyclic cover in a stratum of Abelian differentials in genus four with a maximally degenerate Kontsevich--Zorich spectrum (the only known example in genus three found previously by Forni also corresponds to a square-tiled cyclic cover [15]. We present several new examples of Teichmuller curves in strata of holomorphic and meromorphic quadratic differentials with a maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichmuller curves with maximally degenerate spectra. We prove that this is indeed the case within the class of square-tiled cyclic covers.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6d94cc05894e2201bd339baf235a1d7a06665e5c

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In Geometry and Topology

1. Carlos Matheus
2. G. Forni
3. A. Zorich
4. A. Eskin
5. Martin Möller
6. J. Yoccoz
7. M. Kontsevich
8. Alexander I. Bufetov

In Mathematical Physics

1. A. Zorich
2. Carlos Matheus
3. G. Forni
4. A. Eskin
5. Martin Möller
6. J. Yoccoz
7. M. Kontsevich
8. Marcelo Viana

Citations in the corpus

• Lyapunov spectrum of square-tiled cyclic covers (2010, background): “for k= 1,...,N− 1, form a basis of the space of holomorphic forms. Proof. The statement of the Lemma is equivalent to the following identity: (18) NX−1 k=1 (t(k)− 1) = g. Itis easy tocheck (see, say, [FMZ])thatthe genus gofMN(a1,a2,a3,a4) is expressed in terms of parameters N,ai as g= N+1− 1 2 X4 i=”
• A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces (2013, background): “On the other hand, Forni and his coauthors (see [21,23]) constructed two examples of translation surfaces with the following property: their orbits under the natural SL2(R)-action on moduli space are closed and support SL2(R)invariant probability measures with respect to which all nontrivial Lyapuno”
• Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow (2011, methodology): “ examples of regular PSL(2,R)-invariant suborbifolds M1 in the strata of meromorphic quadratic differentials in genera g≥2 different from the Teichmu¨ller curves of square-tiled cyclic covers listed in [FoMaZ1] having completely degenerate Lyapunov spectrum λ+ 1 = ···= λ+ g = 0 for the bundle H1 +? No”
• Introduction to Teichm\"uller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards (2013, methodology): “e SL(2;R)-invariant probability supported on the locus of (M N(a 1;:::;a 4);!). 39In addition, the Veech group has a simple dependence on N;a 1;:::;a 4 and it has index 1;2;3 or 6 inside SL(2;Z). See [34]. INTRODUCTION TO TEICHMULLER DYNAMICS 95 For \linear algebra reasons", it is better to wor”