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Journal of Lie Theory 17 (2007), No. 4, 857--867
Copyright Heldermann Verlag 2007



A Generalization of Helling-Kim-Mennicke Groups and Manifolds

Elena Barbieri
Dip. di Matematica, UniversitÓ di Modena e Reggio Emilia, Via Campi 213/B, 41100 Modena, Italy

Alberto Cavicchioli
Dip. di Matematica, UniversitÓ di Modena e Reggio Emilia, Via Campi 213/B, 41100 Modena, Italy
alberto.cavicchioli@unimore.it

Fulvia Spaggiari
Dip. di Matematica, UniversitÓ di Modena e Reggio Emilia, Via Campi 213/B, 41100 Modena, Italy
fulvia.spaggiari@unimore.it



We construct an infinite family of closed connected orientable 3-manifolds by pairwise identifications of faces in the boundary of certain polyhedral 3-cells. We determine geometric presentations (that is, induced by Heegaard diagrams of the constructed manifolds) of the fundamental group, and study the split extension of it. Then we prove that these manifolds are n-fold cyclic coverings of the 3-sphere branched over some pretzel links. Our results generalize those of Helling, Kim and Mennicke [Comm. in Algebra 23 (1995) 5169--5206] and Cavicchioli and Paoluzzi [Manuscripta Math. 101 (2000) 457--494] on cyclic branched coverings of the Whitehead link, and their fundamental groups.

Keywords: 3-manifolds, group presentations, spines, orbifolds, polyhedral schemata, branched coverings, hyperbolic structure.

MSC: 57M12, 57M25, 53C30; 20F05, 20F34, 57M50, 57N10

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