5 research outputs found
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 or a direct product of two cyclic groups of order .Comment: 22 pages, AMS-Te
Extensions of the representation modules of a prime order group
For the ring R of integers of a ramified extension of the field of p-adic numbers and a cyclic group G of prime order p we study the extensions of the additive groups of R-representations modules of G by the group G
Torsion-free crystallographic groups with indecomposable holonomy group II.
Let K be a principal ideal domain, G a finite group, and M a KG-module which is a free K-module of finite rank on which G acts faithfully. A generalized crystallographic group is a non-split extension C of M by G such that conjugation in C induces the G-module structure on M. ( When K = Z, these are just the classical crystallographic groups.) The dimension of C is the K-rank of M, the holonomy group of C is G, and C is indecomposable if M is an indecomposable KG-module.
We study indecomposable torsion-free generalized crystallographic groups with holonomy group G when K is Z, or its localization Z((p)) at the prime p, or the ring Z(p) of p-adic integers. We prove that the dimensions of such groups with G non-cyclic of order p(2) are unbounded. For K = Z, we show that there are infinitely many non-isomorphic such groups with G the alternating group of degree 4 and we study the dimensions of such groups with G cyclic of certain orders