| Kleinian Groups Generated by Rotations (2007) | |||||||||||||
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| . We discuss which Kleinian groups are commensurable with Kleinian groups generated by rotations, with particular emphasis on Kleinian groups that arise from Dehn surgery on a knot. Introduction In a problem session, organized by A. Kim, of the 1991 German-Korean-SEAMS Conference on Geometry, E. Vinberg asked for a cocompact Kleinian group which is not commensurable with a group generated by rotations (elements of finite order). Examples are, in fact, not hard to find. In this note we describe in some detail which Kleinian groups have this property among the Kleinian groups that occur as fundamental groups of Dehn surgeries on knots. We also briefly discuss some related questions. In the following \Gamma and will always denote Kleinian groups of finite covolume, that is, discrete subgroups of PSL(2; C ) = Isom + (H 3 ) such that the orbifold H 3 =\Gamma or H 3 = has finite volume. They are commensurable if they have isomorphic subgroups of finite index. By Mostow-Prasad rigi... | |||||||||||||
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