An inequality for polyhedra and ideal triangulations of cusped hyperbolic 3-manifolds

Abstract

It is not known whether every noncompact hyperbolic 3-manifold of finite volume admits a decomposition into ideal tetrahedra. We give a partial solution to this problem: Let M M be a hyperbolic 3-manifold obtained by identifying the faces of n n convex ideal polyhedra P 1 , … , P n P_{1},\dots ,P_{n} . If the faces of P 1 , … , P n − 1 P_{1},\dots ,P_{n-1} are glued to P n P_{n} , then M M can be decomposed into ideal tetrahedra by subdividing the P i P_{i} ’s.