Counting generalized Jenkins-Strebel differentials

Jayadev S. Athreya, Alex Eskin, Anton Zorich

Research output: Contribution to journalArticlepeer-review


We study the combinatorial geometry of "lattice" Jenkins-Strebel differentials with simple zeroes and simple poles on CP1 and of the corresponding counting functions. Developing the results of Kontsevich (Commun Math Phys 147:1-23, 1992) we evaluate the leading term of the symmetric polynomial counting the number of such "lattice" Jenkins-Strebel differentials having all zeroes on a single singular layer. This allows us to express the number of general "lattice" Jenkins-Strebel differentials as an appropriate weighted sum over decorated trees. The problem of counting Jenkins-Strebel differentials is equivalent to the problem of counting pillowcase covers, which serve as integer points in appropriate local coordinates on strata of moduli spaces of meromorphic quadratic differentials. This allows us to relate our counting problem to calculations of volumes of these strata. A very explicit expression for the volume of any stratum of meromorphic quadratic differentials recently obtained by the authors (Athreya et al. 2012) leads to an interesting combinatorial identity for our sums over trees.

Original languageEnglish (US)
Pages (from-to)195-217
Number of pages23
JournalGeometriae Dedicata
Issue number1
StatePublished - Jun 2014


  • Jenkins-Strebel differential
  • Meromorphic quadratic differential
  • Pillowcase covers

ASJC Scopus subject areas

  • Geometry and Topology


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