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

Abstract

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.

25 pages. Complete rewrite

References in corpus (5)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Tameness of hyperbolic 3-manifolds
• Representations of Galois Groups on the Homology of Surfaces
• Residually free 3-manifolds

Cited by in corpus (2)

• On some of the residual properties of finitely generated nilpotent groups
• Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups