By Grzegorz Andrzejczak (auth.), Julian Ławrynowicz (eds.)

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Diff. Geom. ~ (1968), 421-446. : ' The degree of polynomial growth of finitely generated 44 nilpotent groups', Proc. Lond. S. (3) [8] [9] ~~ (1972), 602-614. : 'Degrees of growth of finitely generated groups and the theory of invariant means i , Math. USSR Iz. ~2 (1985), 259-99. : 'Invariant measures and growth conditions', Trans. ~ (1974), 33-53. ArneY'. S. [10] EVANS, B. : 'Solvable fundamental groups of compact 3-manifo1ds', TY'ans. ArneY'. S. 168 (1972), 189-210. : 'Existence of quasiregular mappings from to closed orientable 3-manifolds', to appear.

The star operator *: AP,q+Am- p m-q is defined by the requirement that \jJ(x)l\*n( .. ) = (\jJ(x) n(x) dV(x). By the above it is easy to verify that ** n = (-l)P q n lor every n E AP,q. 1. The fOY'lTlal adjoint a: AP, q 1 + AP , q is given by Proof. 8*: AP,q+A P , q-l of By applying the Stokes theorem we get ~ -a cp,\jJ>= acp/\*lj! ¢, M * a * \jJ > . a *)\)J = The a-Laplacian 1I8: AP , q + ~p,q is defined by 1I3 = a*a + aa*. (p,q)-form \jJ is said to be d pharmonic if lid lj! = O. Let us denote by HE,q = zE,q/B ,q the (p,q)-cohomology group in the sense of Dolbeault.

Our intention is to give an alternative construction which is slightly more intrinsic and comes from integrating cocycles associated with the normal bundle. The existence of such cocycles allows us to modify the construction of [6] in order to make it applicable to transversely holomorphic foliations (and even to a wide class of foliations with an integrable transverse G-structure [2]). 1. 1. For any smooth foliation F of a (real) manifold M lits principal normal bund\f is the manifold of I-jets PF={j ¢; ¢(x) =O} of submersions (to lR) defining F locally.

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