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Hurder, Essential solenoids, ghost cycles, and F Γrq , preprint, 2008. 27. S. Hurder and D. T¨ oben, Trans. Amer. Math. , to appear. 28. S. Hurder and D. T¨ oben, Equivariant basic cohomology and residues of Riemannian foliations, in preparation, 2008. ´ 29. W. Kamber and Ph. Tondeur, Ann. Sci. Ecole Norm. , 8:433–486, 1975. 30. W. Kamber and Ph. Tondeur, Foliated bundles and characteristic February 17, 2009 16:46 WSPC - Proceedings Trim Size: 9in x 6in viii-icdg 35 classes, Lect. Notes in Math.

Let I be a compact interval in R, and fI : R × (0, ∞) → (0, ∞) the function fI (w) = ϕ(w, z, p) dν(z, p) . I×N It is harmonic. Because m ˜ is mapped to V by the projection (w, z, p) → (w, z), we have fI = cst · Im(w). e. (z, p) ϕ(w, z, p) = Im(w) . Recall that ν is mapped by (z, p) → z to the Lebesgue measure dz. By Fubini’s theorem, there is a family of probability measures µz on N , depending measurably on z, such that ν= µz dz . The map (x, z) → µz is invariant by the translations (x, z) → (x+ai , z +bi ), i = 1, 2.

February 17, 2009 16:46 WSPC - Proceedings Trim Size: 9in x 6in viii-icdg 39 • (T2 × (0, ∞), F) the quotient of (H × R, F) by G . • (M, Fq ) the quotient of (H × R, F) by G. |dw| , be a Riemannian metric on the leaves of F: it is invariant Let ds2 = Im(w) under the group G. We denote by ds2 the induced metric on the leaves of F and Fq . Let N be a compact smooth manifold and Φ : N → N be a diffeomorphism. We denote (NΦ , FΦ ) the quotient of the product T2 × (0, ∞) × N , together with the product foliation F × {·}, by the map (g0 , Φ).