Mathematics – Group Theory
Scientific paper
2010-02-17
Mathematics
Group Theory
Some changes made to improve readability and motivation and to correct typographical errors. The content is not changed
Scientific paper
In recent years, the RFRS condition has been used to analyze virtual fibering in 3-manifold topology. Agol's work shows that any 3-manifold with zero Euler characteristic satisfying the RFRS condition on its fundamental group virtually fibers over the circle. In this note we will show that a finitely generated nilpotent group is either virtually abelian or is not virtually RFRS, a result which may be of independent interest though not directly applicable to 3-manifold topology. As a corollary, we deduce that any RFRS group cannot contain a nonabelian torsion-free nilpotent group. This result also illustrates some of the interplay between residual torsion-free nilpotence and the RFRS condition.
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