Word hyperbolic Dehn surgery

The aim of this paper is to demonstrate that very many Dehn fillings on a cusped hyperbolic 3-manifold yield a 3-manifold which is irreducible, atoroidal and not Seifert fibred, and which has infinite, word hyperbolic fundamental group. We establish an extension of the Thurston-Gromov $2\pi$ theorem by showing that if each filling slope has length more than six, then the resulting 3-manifold has all the above properties. We also give a combinatorial version of the $2\pi$ theorem which relates to angled ideal triangulations. We apply these techniques by studying surgery along alternating links.