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Flat manifolds with only finitely many affinities. (English) Zbl 0894.57026

The group of affinities \(\text{Aff}(X)\) of a closed flat (Riemannian) manifold \(X\) is determined by an exact sequence \(1\to \text{Aff}_0(X)\to \text{Aff}(X)\to \text{Out}(\pi_1 X)\to 1\) where \(\text{Aff}_0(X)\) denotes the identity component of \(\text{Aff}(X)\) (a torus whose dimension is the first Betti number of \(X\)) and \(\text{Out}(\pi_1 X)\) the outer automorphism group of the fundamental group of \(X\) (which in turn is determined by the finite holonomy of \(X\)). The main object of the present paper are flat manifolds with finite, and in particular small groups of affinities, with a special regard to the following natural and interesting questions: which finite groups occur as the affinity groups of flat manifolds; in particular, is there a flat manifold with trivial group of affinities? This remains open in general but the main point of the present paper is the construction of flat manifolds \(X\) such that \(\text{Aff}(X)\cong\mathbb{Z}_2\) has only two elements. Also, the finite groups of affinities of certain classes of generalized Hantzsche-Wendt manifolds are computed.

MSC:

57S30 Discontinuous groups of transformations
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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