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BRANCHED SURFACES AND THE SIMPLEST FOLIATIONS OF 3-MANIFOLDS

The set of foliations carried by the same branched surface as a foliation f includes, but is not restricted to, the set of foliations sufficiently close to f. We show that a branched surface may be constructed to ensure that no foliation carried by it has more dead-end components than f. In particular, if f is taut, all foliations carried by the branched surface will be taut. We use the result to classify the foliations of closed 3manifolds having the fewest number of dead-end components. We then develop an algorithm to determine a lower bound on the depth of a foliation in a given class.