Abstract
A geometric orbifold of dimension d is the quotient space S = X/K, where (X,G) is a geometry of dimension d and K < G is a co-compact discrete subgroup. In this case {ie38-01} is called the orbifold fundamental group of S. In general, the derived subgroup K’ of K may have elements acting with fixed points; i.e., it may happen that the homology cover MS = X/K’ of S is not a geometric manifold: it may have geometric singular points. We are concerned with the problem of deciding when K′ acts freely on X; i.e., when the homology cover M S is a geometric manifold. In the case d = 2 a complete answer is due to C. Maclachlan. In this paper we provide necessary and sufficient conditions for the derived subgroup S to act freely in the case d = 3 under the assumption that the underlying topological space of the orbifold K is the 3-sphere S 3.
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Original Russian Text Copyright © 2010 Hidalgo R. A. and Mednykh A. D.
The authors were partially supported by Fondecyt (Grants 7050189, 1060378, and 1070271), the UTFSM 12.08.01, and the Russian Foundation for Basic Research (Grant 09-01-00255).
Valparaiso; Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 1, pp. 48–61, January–February, 2010.