Minimal dimensions for flat manifolds with prescribed holonomy (Q1105238)

The paper develops some general methods to find the minimal dimensions of compact connected flat Riemannian manifolds with given isomorphism class for its holonomy group or equivalently the minimal dimension of a fixed point free crystallographic space group with given isomorphism type for its point group. The methods are based on the investigation of the p-adic completions of the space groups involved and are made quite explicit for point groups having cyclic Sylow subgroups for some primes. Since the method can be used for many perfect groups it extends the range of earlier work by various authors on the problem for soluble groups. The results are applied to find these minimal dimensions for \(PSL_ 2(p)\) and \(SL_ 2(p)\).