Seven dimensional flat manifolds with cyclic holonomy

Profinite genus of fundamental groups of compact flat manifolds with the cyclic holonomy group of square-free order

In this article we study the extent to which an $n$-dimensional compact flat manifold with the cyclic holonomy group of square-free order may be distinguished by the finite quotients of its fundamental group. In particular, we display a formula for the cardinality of profinite genus of the fundamental group of an $n$-dimensional compact flat manifold with the cyclic holonomy group of square-free order.

On cohomology of crystallographic groups with cyclic holonomy of split type

We disprove a conjecture stating that the integral cohomology of any crystallographic group Z^n \rtimes Z_m is given by the cohomology of Z_m with coefficients in the cohomology of the group Z^n, by providing a complete list of counterexamples up to dimension 5. We also find a counterexample with odd order holonomy, m=9, in dimension 8 and finish the computations of the cohomology of 6-dimensional crystallographic groups arising as orbifold fundamental groups of certain Calabi-Yau toroidal orbifolds.

Torsion free groups with indecomposable holonomy group I

We study the torsion free generalized crystallographic groups with the indecomposable holonomy group which is isomorphic to either a cyclic group of order ${p^s}$ or a direct product of two cyclic groups of order ${p}$.

Holonomy groups of compact flat solvmanifolds

This article is concerned with the study of the holonomy group of flat solvmanifolds. It is known that the holonomy group of a flat solvmanifold is abelian; we give an elementary proof of this fact and moreover we prove that any finite abelian group is the holonomy group of a flat solvmanifold. Furthermore, we show that the minimal dimension of a flat solvmanifold with holonomy group $\mathbb{Z}_n$ coincides with the minimal dimension of a compact flat manifold with holonomy group $\mathbb{Z}_n$. Finally, we give the possible holonomy groups of flat solvmanifolds in dimensions 3, 4, 5 and 6; e…

Generators of Bieberbach groups with 2-generated holonomy group

An n-dimensional Bieberbach group is the fundamental group of a closed flat $n$-dimensional manifold. K. Dekimpe and P. Penninckx conjectured that an n-dimensional Bieberbach group can be generated by n elements. In this paper, we show that the conjecture is true if the holonomy group is 2-generated (e.g. dihedral group, quaternion group or simple group) or the order of holonomy group is not divisible by 2 or 3. In order to prove this, we show that an n-dimensional Bieberbach group with cyclic holonomy group of order larger than two can be generated by n-1 elements.

源自 Artin 辫群的 Bieberbach 群与具有有限阿贝尔和乐的平坦流形

Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups

本文证明对于 $S_n$ 的任意有限阿贝尔子群，晶体群 $B_n/[P_n,P_n]$ 包含相应的 Bieberbach 子群，给出了其和乐表示的显式描述，并以此研究了奇数阶循环和乐群下平坦流形的 Anosov 微分同胚存在性与 Kähler 几何性质。