# Virtual Finite Quotients of Finitely Generated Groups

### From NZJM

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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.

**Abstract** If *G* is a semidirect product *N* *H* 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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