Profinite rigidity in low dimensions, III
1. Profinite rigidity of classes of closed 3-manifolds
We are interested in the restricted genus
Theorem 1 (Wilton-Zalesskii) Let , be the fundamental groups of closed 3-manifolds, with .
- If one is hyperbolic, so is the other.
- If one is a Seifert fibered space, so is the other.
For (1), the point is to detect ‘s and free products. Free product decompositions are detected by the first -Betti number. For ‘s, the argument relies on Agol-Wise theory.
For (2), note that there exist non-diffeomorphic Seifert fiber spaces with isomorphic profinite completions.
Garret Wilkes recently proved that Hempel’s is the only possible construction.
Funar showed the existence of non-diffeomorphic Sol manifolds with isomorphic profinite completions. They correspond to explicit matrices which are not conjugate in but are conjugate in any congruence quotient.
1.1. Rigidity for fibering 3-manifolds
There is a link complement which surjects onto all finite simple groups, and has the same collection of lower central series quotients as .
Theorem 2 (Bridson-Reid-Wilton) Let , be the fundamental groups of finite volume hyperbolic 3-manifolds, with .
- If one is fibered with , so is the other, with the same genus fiber (closed case) or rank of fiber free group (noncompact case).
- If is a 1-punctured torus bundle, then .
Boileau-Friedl had a special case of (2), for the figure 8 knot complement.
1.2. Aside on LERF
Suppose . We get a topology induced from the profinite topology of , and can consider completion , with a continuous surjection . When is this map injective ?
Example. Answer is positive for finite index subgroups.
Exercise. Say is -separable if is closed in the profinite topology on . I.e. is the intersection of finite index subgroups of that contain it. Say is LERF if is -separable for all finitely generated subgroups . Show that above question has a positive answer if is LERF.
Theorem 3 (Agol, Wise, Scott for Seifert fiber spaces)
- Geometric closed 3-manifold groups are LERF.
- Non-compact finite volume hyperbolic manifold groups and non-compact Seifert fibered space groups are LERF.
The converse of (1) is true (Hongbin Sun).
Non-example. For and be a corner , answer is no. This follows from the Congruence subgroup property.
Once fibering is established, the genus and rank statement follows easily. Indeed, thanks to LERF, exact sequences yield exact sequences of profinite completions. makes the exact sequence unique, so the kernel is uniquely defined.
For 1-punctured torus bundles, there are only 2 possibilities for the fiber. One shows that a triply punctured sphere fiber arises only for Seifert fibered spaces. Hyperbolic 1-punctured torus bundles have monodromy a hyperbolic element of (Jorgensen). The cardinality of the torsion part of is , so this leaves only finitely many possibilities.
2.1. Proof of BRW
Assume that is finite volume non-compact hyperbolic. Let be profinite equivalent to . Start with a few reductions.
Assume is compact with a single incompressible torus boundary component. Task: show that is fibered.
Assume that is finite volume hyperbolic. implies that surjects onto with kernel . Freedman shows that contains a closed surface group . Wise shows that is LERF, so exact sequences go to profinite completions.
Beware that if is a free group, contains dense surface groups (Breuillard-Gelander-Souto-Storm). So one must take care of closedness of subgroups. There is a notion of cohomological dimension for profinite groups. Closed surface subgroups contribute, but not dense surface subgroups.