Torsion-free crystallographic groups with indecomposable holonomy group II.

Bódi, Viktor and Gudivok, P. M. and Rudko, V. P. (2004) Torsion-free crystallographic groups with indecomposable holonomy group II. Journal of Group Theory, 7 (4). pp. 555-569. ISSN 1433-5883 (print), 1435-4446 (online)

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Abstract

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.

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