manuscripta math. 90, 383 - 389 (1996) manuscripta mathematlca C Spdnger-Y 9 8 1996 Holonomy groups of five dimensional Bieberbach groups Andrzej Szczepafiski 1 Institute of Mathematics, University of Gdafisk, ul.Wita Stwosza 57, 80-952 Gdafisk, Poland, e-mail : matas@paula.univ.gda.pl Received March 18, 1996 Let F be a Bieberbach group - that is a torsion fl'ee crystallographic group. Ill this paper we give a list of the isomorphism types of all holonomy groups of five- dimensional Bieberbach groups. An application to the problem of estimating the covolume of a discrete group of orientation-preserving isometries of hyperbolic 6-space is also given. 1 Introduction One of the purpose of this paper is to answer the question: Which finite groups can be holonomy groups of 5-dimensional Bieberbach groups? For 5-dimensional Bieberbach groups with trivial centre the answer for tile above question was given in [8], Hence we need only consider the ease of Bieber- bach groups with non-trivial centre. We assume familiarity with [8] and retain the notations and definitions given there. We would also like to say that our methods in principle use the Calabi reduction (see [7] pp. 9-11) although we do not always mention it. Another purpuse of this note is to give a new estimate for the covolume of a discrete group of orientation-preserving isometries of hyperbolic 6-space. This is done with the support of Theorem 1 which gives us the value of/5 (for the definition see the last part). 1This work was supported by Polish grant nr.0627/P3/93/04