Cohomology Classes Represented by Measured Foliations, and Mahler’s Question for Interval Exchanges

A translation surface on (S,Σ) gives rise to two transverse measured foliations F ,G on S with singularities in Σ, and by integration, to a pair of cohomology classes [F ], [G] ∈ H(S,Σ;R). Given a measured foliation F , we characterize the set of cohomology classes b for which there is a measured foliation G as above with b = [G]. This extends previous results of Thurston [Th] and Sullivan [Su]. We apply this to two problems: unique ergodicity of interval exchanges and flows on the moduli space of translation surfaces. For a fixed permutation σ ∈ Sd, the space R d + parametrizes the interval exchanges on d intervals with permutation σ. We describe lines l in Rd+ such that almost every point in l is uniquely ergodic. We also show that for σ(i) = d+1− i, for almost every s > 0, the interval exchange transformation corresponding to σ and (s, s, . . . , s) is uniquely ergodic. As another application we show that when k = |Σ| ≥ 2, the operation of ‘moving the singularities horizontally’ is globally well-defined. We prove that there is a well-defined action of the group B ⋉ R on the set of translation surfaces of type (S,Σ) without horizontal saddle connections. Here B ⊂ SL(2,R) is the subgroup of upper triangular matrices.