Free actions of finite Abelian groups on the 3-torus

Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group

“…Not including the torus itself, there are exactly 3 possibilities for a manifold double covered by the 3-torus [20]. Each is defined by taking a quotient of T 3 = R 3 /Z 3 by one of the following maps:…”

K-theory of noncommutative Bieberbach manifolds

“…The result itself is not entirely surprising: the K 0 groups have torsion, although the exact form of the torsion component as well as the fact that n the Z 6 case there is no torsion cannot be seen at once. It is a remarkable fact that there exists a striking relation between the K 0 groups of the manifolds BN (N = 2, 3, 4, 6) and the first homology groups of the corresponding infinite Bieberbach groups G N [9], (so that BN = R 3 /G N ), namely K 0 (BN) ∼ Z ⊕ H 1 (G N , Z). This fact, as well the remaining case of nonorientable manifolds, together with the study of spectral geometries and spin structures over noncommutative Bieberbach manifolds shall be discussed in our future work.…”

“…The above actions give rise to five oriented name group G generators of G action of G on U, V, W B2 Z 2 e e U = −U , e V = V * , e W = W * B3 Z 3 e e U = e 2 3 πi U , e V = W * , e W = W * V B4 Z 4 e e U = iU , e V = W , e W = V * B5 Z 2 × Z 2 e 1 , e 2 e 1 U = −U , e 1 V = V * , e 1 W = W * e 2 U = U * , e 2 V = −V , e 2 W = −W * B6 Z 6 e e U = e 1 3 πi U , e V = W , e W = W V * Table 1: Orientable actions of finite groups on three-torus flat three-manifolds different from the torus. The remaining four nonorientable quotients, originate from the following actions: For full details and classifications of all free actions of finite groups on threetorus see [9,10], note, however, that the resulting quotient manifolds are always one of the above Bieberbach manifolds. It is easy to see that N1 is just the Cartesian product of S 1 with the Klein bottle, whereas N3 and N4 are two distinct Z 2 quotients of B2.…”

“…The action of the finite groups on the unitaries U, V, W , which generate the algebra of continuous functions on the three-torus is presented in the table below and comes directly from the action of Bieberbach groups on R 3 Table 1: Orientable actions of finite groups on three-torus flat three-manifolds different from the torus. The remaining four nonorientable quotients, originate from the following actions: For full details and classifications of all free actions of finite groups on threetorus see [9,10], note, however, that the resulting quotient manifolds are always one of the above Bieberbach manifolds. It is easy to see that N1 is just the Cartesian product of S 1 with the Klein bottle, whereas N3 and N4 are two distinct Z 2 quotients of B2.…”

Sets of Periods for Expanding Maps on Flat Manifolds

“…(vii) ( [8]) if ϕ : Γ → Γ is an injective homomorphism of the affine Bieberbach group Γ , there exists (A, a) ∈ Z Z n×n < IR n ⊂ Aff(IR n ) such that, for all γ = (U, u) ∈ Γ :…”