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In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most 1, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a negative answer to Thurston’s conjecture. Here counter-examples have been constructed conditional on the fully marked surface theorem. In the second paper, joint with David Gabai, a proof of the fully marked surface theorem is given.

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An axiomatic formulation is presented for point processes which may be interpreted as ordered sequences of points randomly located on the real line. Such concepts as forward recurrence times and number of points in intervals are defined and related in set-theoretic Note that for α∈$A$,$G$^{α}may not cover$G$_{α}as a convex subgroup and so we cannot use Theorem 1.1 to prove this result. Moreover, all that we know about the$G$^{α}/G_{α}is that each is an extension of a trivially ordered subgroup by a subgroup of$R$. It$B$is a plenary subset of$A$, then there exists a$v$-isomorphism μ of$G$into$V(B, G$^{β}/G_{β}), but whether or not μ is an$o$-isomorphism is not known.

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