Canonical triangulations of Dehn fillings

Abstract

Every cusped, finite-volume hyperbolic three-manifold has a canonical decomposition into ideal polyhedra. We study the canonical decomposition of the hyperbolic manifold obtained by filling some (but not all) of the cusps with solid tori: in a broad range of cases, generic in an appropriate sense, this decomposition can be predicted from that of the unfilled manifold (a similar result has been independently announced by Akiyoshi [Kokyuroku 1329, RIMS, Kyoto (2003) 121-132]). We also find the canonical decompositions of all hyperbolic Dehn fillings on one cusp of the Whitehead link complement.