Profinite rigidity and surface bundles over the circle
Bulletin of the London Mathematical Society
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Wilton, H., Bridson, M., & Reid, A. (2017). Profinite rigidity and surface bundles over the circle. Bulletin of the London Mathematical Society https://doi.org/10.1112/blms.12076
If $M$ is a compact 3-manifold whose first betti number is 1, and $N$ is a compact 3-manifold such that $π_1N$ and $π_1M$ have the same finite quotients, then $M$ fibres over the circle if and only if $N$ does. We prove that groups of the form $F_2$ $\rtimes$ $\Bbb Z$ 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 $π_1M$ and $π_1N$ maps onto $G$ and the other does not.
The first author was supported in part by grants from the EPSRC and a Royal Society Wolfson Merit Award. The second author was supported in part by an NSF grant and The Wolfensohn Fund. He would also like to thank the Institute for Advanced Study for its hospitality whilst this work was completed. The third author was supported by a grant from the EPSRC.
Engineering and Physical Sciences Research Council (EP/L026481/1)
External DOI: https://doi.org/10.1112/blms.12076
This record's URL: https://www.repository.cam.ac.uk/handle/1810/265747