Profinite rigidity and surface bundles over the circle

Abstract

If $M$ is a compact 3-manifold whose first betti number is 1, and $N$ is a compact 3-manifold such that ${\pi }_{1}N$ and ${\pi }_{1}M$ have the same finite quotients, then $M$ fibres over the circle if and only if $N$ does. We prove that groups of the form $F_2⋊\mathbb{Z}$ are distinguished from one another by their profinite completions. Thus, regardless of betti number, if $M$ and $N$ are punctured torus bundles over the circle and $M$ is not homeomorphic to $N$, then there is a finite group $G$ such that one of ${\pi }_{1}M$ and ${\pi }_{1}N$ maps onto $G$ and the other does not.

17 pages, no figures. v2 minor corrections. This is the final version accepted for publication

References in corpus (3)

• Knot Floer homology detects fibred knots
• Some 3-manifold groups with the same finite quotients
• The profinite completion of $3$-manifold groups, fiberedness and the Thurston norm

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• Profinite rigidity in the SnapPea census
• Profinite rigidity for free-by-cyclic groups with centre
• Mapping classes are almost determined by their finite quotient actions
• Prosoluble subgroups of the profinite completion of the fundamental group of compact 3-manifolds
• A hyperbolic free-by-cyclic group determined by its finite quotients
• Finite quotients, arithmetic invariants, and hyperbolic volume