Virtual Finite Quotients of Finitely Generated Groups

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New Zealand Journal of Mathematics

Vol. 41, (2011), Pages 1-15


J. O. Button

Selwyn College

University of Cambridge

Cambridge CB3 9DQ

U.K.

mailto:jb128@dpmms.cam.ac.uk






Abstract If G is a semidirect product N Image:semidirectproduct.pngH with N finitely generated then G has the property that every finite group is a quotient of some finite index subgroup of G if and only if one of N and H has this property. This has applications to 3-manifolds; for instance for any fibred hyperbolic 3-manifold M and any finite simple group S, there is a finite cyclic cover of M whose fundamental group surjects to S. We also give a short proof of the residual finiteness of ascending HNN extensions of finite rank free groups when the induced map on homology is injective.

Keywords

Classification (MSC2000) Primary 20E07; Secondary 20E26..

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