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While thinking of whether any web (spatial trivalent graph) without an embedded bridge can be realized as a branching locus of a finite branched cover over $S^3$, I realized that this problem is closely related to whether fundamental groups of web complements are residually finite or not. I heard that knot groups are residually finite. Are fundamental groups of web complements also residually finite?

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    $\begingroup$ All finitely generated $\pi_1$ of 3-manifolds are residually finite. $\endgroup$ – YCor Jun 3 '18 at 21:47
  • $\begingroup$ @YCor Thanks for the comment. Could you please provide a reference? $\endgroup$ – Henry Jun 3 '18 at 21:50
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    $\begingroup$ Even better, the complement of a web in a closed 3-manifold deformation retracts onto a compact 3-manifold with boundary (obtained by deleting a small regular neighborhood of the web; in a neighborhood of a vertex this regular neighborhood looks like a "solid pair of pants"). So you only need the argument in this case. YCor's generalization follows immediately from Scott's "compact core" theorem. My memory is hazy but I believe the boundary case may be proved by reduction to the case of incompressible boundary, and then passing through the double. In any case, should be an easier search. $\endgroup$ – mme Jun 3 '18 at 21:59
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    $\begingroup$ Have a look at the book, 3-manifold Groups by Henry Wilton, Matthias Aschenbrenner, and Stefan Friedl. It has a wealth of information on this and similar questions. $\endgroup$ – Danny Ruberman Jun 3 '18 at 22:37
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That the fundamental groups of compact $3$-manifolds are residually finite was proved by Hempel in

Hempel, John, Residual finiteness for 3-manifolds. Combinatorial group theory and topology (Alta, Utah, 1984), 379–396, Ann. of Math. Stud., 111, Princeton Univ. Press, Princeton, NJ, 1987.

Hempel's proof required geometrization (which of course was not known at the time, but is now). As far as I know, there is no proof that does not depend on geometrization.

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  • $\begingroup$ Well, it sounds like this question only needs geometrisation for Haken 3-manifolds, which was known (at least to Thurston) at the time. $\endgroup$ – HJRW Jun 5 '18 at 9:43

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