  In 1976, Thurston proved that taut foliations on closed hyperbolic
3-manifolds have Euler class of norm at most one, 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
counterexamples 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.
