Residual properties of 3-manifold groups I: Fibered and hyperbolic 3-manifolds

Details Residual properties of 3-manifold groups I: Fibered and hyperbolic 3-manifolds Residual properties of 3-manifold groups I: Fibered and hyperbolic 3-manifolds

2009-10-11

arxiv.org/abs/0910.2035v2

Mathematics

Geometric Topology

25 pages. Complete rewrite

Scientific paper

Let $p$ be a prime. In this paper, we classify the geometric 3-manifolds whose fundamental groups are virtually residually $p$. Let $M=M^3$ be a virtually fibered 3-manifold. It is well-known that $G=\pi_1(M)$ is residually solvable and even residually finite solvable. We prove that $G$ is always virtually residually $p$. Using recent work of Wise, we prove that every hyperbolic 3-manifold is either closed or virtually fibered and hence has a virtually residually $p$ fundamental group. We give some generalizations to pro-$p$ completions of groups, mapping class groups, residually torsion-free nilpotent 3-manifold groups and central extensions of residually $p$ groups.

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