A Flat Manifold with No Symmetries

where G is a finite group and L is a faithful ZG-lattice of finite rank, i.e., a free Z-module of finite rank on which G acts faithfully. Let X be a flat manifold with fundamental group Γ. The group Aff(X) of affine self-equivalences of X is a Lie group. Its identity component Aff0(X) is a torus whose dimension is the rank of the center of Γ, and Aff(X)/Aff0(X) is isomorphic to Out(Γ), the outer automorphism group of Γ. Malfait conjectured [Malfait 98, Conjecture 5.13], that Aff(X) is never torsion-free (where the trivial group is considered to be torsion-free). In Section 2, we will give an example of a Bieberbach group that has a trivial center and trivial outer automorphism group, and hence is the fundamental group of a flat manifold with trivial group of affinities. In particular, it is a counterexample to Malfait’s conjecture. Let Γ be a Bieberbach group as in (1—1) and δ ∈ H(G,L) be the cohomology class giving rise to (1—1). Let N be the normalizer of G in Aut(L). There is a natural action of N on H(G,L), and Out(Γ) satisfies the short exact sequence (see [Charlap 86, Theorem V.1.1])