Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 020, 17 pages      arXiv:0902.2977      https://doi.org/10.3842/SIGMA.2009.020

Discrete Cocompact Subgroups of the Five-Dimensional Connected and Simply Connected Nilpotent Lie Groups

Amira Ghorbel and Hatem Hamrouni
Department of Mathematics, Faculty of Sciences at Sfax, Route Soukra, B.P. 1171, 3000 Sfax, Tunisia

Received July 16, 2008, in final form February 09, 2009; Published online February 17, 2009

Abstract
The discrete cocompact subgroups of the five-dimensional connected, simply connected nilpotent Lie groups are determined up to isomorphism. Moreover, we prove if G = N × A is a connected, simply connected, nilpotent Lie group with an Abelian factor A, then every uniform subgroup of G is the direct product of a uniform subgroup of N and Zr where r = dim A.

Key words: nilpotent Lie group; discrete subgroup; nil-manifold; rational structures, Smith normal form; Hermite normal form.

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