Virtual finite quotients of finitely generated groups

TL;DR

This paper investigates conditions under which finitely generated groups have all finite groups as quotients of their finite index subgroups, with applications to 3-manifolds and cyclically presented groups, and provides a new proof of residual finiteness for certain HNN extensions.

Contribution

It characterizes when a semidirect product group has all finite groups as quotients and offers a new proof of residual finiteness for ascending HNN extensions of free groups.

Findings

A semidirect product G has all finite groups as quotients iff either N or H does.

Any fibred hyperbolic 3-manifold has a cyclic cover with fundamental group surjecting onto any finite simple group.

Residual finiteness of ascending HNN extensions of free groups when the induced homology map is injective.

Abstract

If G is a semidirect product N by H with N normal and 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 and to cyclically presented groups, for instance for any fibred hyperbolic 3-manifold M and any finite simple group S, there is a 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.