Geodesic Ideal Triangulations Exist Virtually

It is shown that every non-compact hyperbolic manifold of finite volume has a finite cover admitting a geodesic ideal triangulation. Also, every hyperbolic manifold of finite volume with non-empty, totally geodesic boundary has a finite regular cover which has a geodesic partially truncated triangulation. The proofs use an extension of a result due to Long and Niblo concerning the separability of peripheral subgroups. Epstein and Penner [2] used a convex hull construction in Lorentzian space to show that every non-compact hyperbolic manifold of finite volume has a canonical subdivision into convex geodesic polyhedra all of whose vertices lie on the sphere at infinity of hyperbolic space. In general, one cannot expect to further subdivide these polyhedra into ideal geodesic simplices such that the result is an ideal triangulation. That this is possible after lifting the cell decomposition to an appropriate finite cover is the first main result of this paper. A cell decomposition of a hyperbolic n– manifold into ideal geodesic n–simplices all of which are embedded will be referred to as an embedded geodesic ideal triangulation. Theorem 1. Any non-compact hyperbolic manifold of finite volume has a finite regular cover which admits an embedded geodesic ideal triangulation. The study of geodesic ideal triangulations of hyperbolic 3–manifolds goes back to Thurston [13]. They are known to have nice properties through, for instance, work by Neumann and Zagier [10] and Choi [1]. Petronio and Porti [11] discuss the question of whether every non-compact hyperbolic 3–manifold of finite volume has a geodesic ideal triangulation. This question still remains unanswered. Kojima [6] extended the construction by Epstein and Penner to obtain a canonical decomposition into partially truncated polyhedra of any hyperbolic manifold with totally geodesic boundary components. A cell decomposition of a hyperbolic n–manifold with totally geodesic boundary into geodesic partially truncated n– simplices all of which are embedded will be referred to as an embedded geodesic partially truncated triangulation. Received by the editors April 9, 2007. 2000 Mathematics Subject Classification. Primary 57N10, 57N15; Secondary 20H10, 22E40, 51M10.