Tightness and efficiency of irreducible automorphisms of handlebodies II
  We finish proving that an irreducible automorphism f of a handlebody is efficient if, and only if, a certain standard pair of dual f--invariant laminations have the geometric tightness property. In a previous paper it was proved that this tightness property implies efficiency. We now prove the converse. 
Leonardo N. Carvalho
math.GT