Math. Nachr. zyxwvuts 172 (1995) 283-290 zyxwvut About the Possibilities of Application of Relative Hyperbolization of Polyhedra By ANDRZEJ SZCZEPANSKI zyxwvut ') of Gdansk (Received September 29, 1993) 1. Introduction A closed connected zyxwvuts C" Riemannian manifold zyxwv M" is said to be flat if its sectional curvatures are all zero. In [8] HAMRICK and ROYSTER proved that any flat manifold is a boundary (in the topological meaning), i.e., that there exists a C" compact manifold Wn+l such that zy awn'' = M". The above results suggest a natural question about the structure ofthe manifold W"' '. Let us recall the following construction: Let V be a complete Riemannian manifold of constant negative curvature with finite volume (for short, a hyperbolic manifold) and dimension n zyxwvutsrq 2 3. Then V has only finitely many ends El, E,, ..., E, and each zy Ei may be described as follows: Ei = Fi x (0, GO), where F,, with the induced length metric, is a flat manifold and dim Fi = n - 1. Let us consider the manifold k V' = \ (Fi x (si, zyxwvuts 00)) 9 i= 1 where si zyxwvut E (0, GO). The interior of the manifold V' is diffeomorphic to V and k av' = zyxwv u Fi i- 1 carries a flat metric. Conjecture. For any flat manifold M", there exists a hyperbolic manifold W such that one of the ends of W is homeomorphic to M" x (0, co). Here are some equivalent formulations: (1) If r is a Bieberach group of rank n, then r is a maximal subgroup of r' where r' is (2) If M" is a flat Riemannian manifold, then M" = a W"' where W"' \ a W"' supports a fundamental group of a hyperbolic manifold of dimension n + 1. a hyperbolic structure. ') This research was partially supported by Polish grant No. 2 1127901.