By Marco Brunella

The textual content provides the birational type of holomorphic foliations of surfaces. It discusses at size the speculation built via L.G. Mendes, M. McQuillan and the writer to review foliations of surfaces within the spirit of the category of complicated algebraic surfaces.

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**Sample text**

This material is classical. It is collected here for the sake of completeness and to fix notation. 1 We recall that PG = P SL2(M) = SL2(M)/{±1} is the group Aut X of conformal transformations of X. The subgroup #7{±1} is the isotropy subgroup of i:gi = i <<=>> g e K. i yields a diffeomorphism TC : NA = X = G/K, which maps 'l 0 x\ l)'{ (yW 0 0 onto x + yi (x, y e R, y > 0). 2 The action of G on X extends continuously t o X U R U o o . We let X = XUEUoo, which is diffeomorphic to the closed unit disc.

Then pm_x = Yn = 0 and we have (7) £ > , + Ar(Dj)) = n + m—\ m Assume now that the chain is infinite - say, that n has no maximum - and let n -> oo. The left-hand side of (6) is bounded by In + Ar(£2); hence the series on the right-hand side converges and so Yj also has a limit, say Yoo- We claim that Yoo ^ n/2- The sequence of distances d(zo, dj) (j = 0 , 1 , . . ) diverges, so there exist arbitrarily large j such that d(zo,aj+i) >d(zo,aj). This implies that fij > Yj in the triangle Dj and, since fij + Yj ^ TT, yields Yj ^ TT/2, whence our assertion.

Assume first / to be of type n. Denote by V the Downloaded from University Publishing Online. 250 on Tue Jan 24 03:47:54 GMT 2012. 16 SL2(M), differential operators, and convolution 23 smallest G-invariant closed subspace of C°°(G) containing / . Then it can be shown that its elements are annihilated by P(C) and that Vx = { v e V | v is of type x } is finite dimensional for every x £ K. This is the result we need. 21. 12). We see therefore that there exists afinitedimensional subspace L of C°°(G) containing / , stable under the convolutions *a (a e 7C°°(G)) and containing therefore all elements / * a.