Degenerations of quadratic differentials on ℂℙ1

Boissy, Corentin¹

¹ IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France

Geometry & topology, Tome 12 (2008) no. 3, pp. 1345-1386.

Résumé

We describe the connected components of the complement of a natural “diagonal” of real codimension $1$ in a stratum of quadratic differentials on ${ℂ\mathbb{P}}^1$. We establish a natural bijection between the set of these connected components and the set of generic configurations that appear on such “flat spheres”. We also prove that the stratum has only one topological end. Finally, we elaborate a necessary toolkit destined to evaluation of the Siegel–Veech constants.

Détail

DOI : 10.2140/gt.2008.12.1345

Keywords: quadratic differentials, saddle connections, Siegel–Veech constants

Bibliographie

[1] C Boissy, Configuration of saddle connections of quadratic differential on $\mathbb{CP}^1$ and on hyperelliptic Riemann surfaces (2008)

[2] C Boissy, E Lanneau, Dynamics and geometry of the Rauzy–Veech induction for quadratic differentials (2007)

[3] C Danthony, A Nogueira, Measured foliations on nonorientable surfaces, Ann. Sci. École Norm. Sup. $(4)$ 23 (1990) 469

[4] A Eskin, H Masur, A Zorich, Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. Hautes Études Sci. 97 (2003) 61

[5] J Hubbard, H Masur, Quadratic differentials and foliations, Acta Math. 142 (1979) 221

[6] B Hughes, A Ranicki, Ends of complexes, Cambridge Tracts in Mathematics 123, Cambridge University Press (1996)

[7] M Kontsevich, A Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003) 631

[8] H Masur, Interval exchange transformations and measured foliations, Ann. of Math. $(2)$ 115 (1982) 169

[9] H Masur, J Smillie, Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math. $(2)$ 134 (1991) 455

[10] H Masur, S Tabachnikov, Rational billiards and flat structures, from: "Handbook of dynamical systems, Vol. 1A", North-Holland (2002) 1015

[11] H Masur, A Zorich, Multiple saddle connections on flat surfaces and the principal boundary of the moduli space of quadratic differentials

[12] W A Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. $(2)$ 115 (1982) 201

[13] W A Veech, Geometric realizations of hyperelliptic curves, from: "Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992)", Plenum (1995) 217

[14] J C Yoccoz, Continued fraction algorithms for interval exchange maps: an introduction, from: "Frontiers in number theory, physics, and geometry. I", Springer (2006) 401

[15] A Zorich, Flat surfaces, from: "Frontiers in number theory, physics, and geometry. I", Springer (2006) 437