## Notes of Alan Reid’s third Oxford lecture 24-03-2017

Profinite rigidity in low dimensions, III

1. Profinite rigidity of classes of closed 3-manifolds

We are interested in the restricted genus

$\displaystyle \begin{array}{rcl} \mathcal{G}_3(\Gamma)=\{\Delta\, 3-\textrm{compact manifold group}\,\hat\Delta=\hat\Gamma\}. \end{array}$

Theorem 1 (Wilton-Zalesskii) Let ${\Gamma}$, ${\Delta}$ be the fundamental groups of closed 3-manifolds, with ${\hat\Delta=\hat\Gamma}$.

1. If one is hyperbolic, so is the other.
2. If one is a Seifert fibered space, so is the other.

For (1), the point is to detect ${{\mathbb Z}\oplus{\mathbb Z}}$‘s and free products. Free product decompositions are detected by the first ${\ell^2}$-Betti number. For ${{\mathbb Z}\oplus{\mathbb Z}}$‘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 ${Sl_2({\mathbb Z})}$ 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 ${F_2}$.

Theorem 2 (Bridson-Reid-Wilton) Let ${\Gamma}$, ${\Delta}$ be the fundamental groups of finite volume hyperbolic 3-manifolds, with ${\hat\Delta=\hat\Gamma}$.

1. If one is fibered with ${b_1=1}$, so is the other, with the same genus fiber (closed case) or rank of fiber free group (noncompact case).
2. If ${\Gamma}$ is a 1-punctured torus bundle, then ${\mathcal{G}_3(\Gamma)=\{\Gamma\}}$.

Boileau-Friedl had a special case of (2), for the figure 8 knot complement.

1.2. Aside on LERF

Suppose ${H<\Gamma}$. We get a topology induced from the profinite topology of ${\hat\Gamma}$, and can consider completion ${\bar H}$, with a continuous surjection ${\hat H\rightarrow \bar H}$. When is this map injective ?

Example. Answer is positive for finite index subgroups.

Exercise. Say ${\Gamma}$ is ${H}$-separable if ${H}$ is closed in the profinite topology on ${\Gamma}$. I.e. ${H}$ is the intersection of finite index subgroups of ${\Gamma}$ that contain it. Say ${\Gamma}$ is LERF if ${\Gamma}$ is ${H}$-separable for all finitely generated subgroups ${H<\Gamma}$. Show that above question has a positive answer if ${\Gamma}$ is LERF.

Theorem 3 (Agol, Wise, Scott for Seifert fiber spaces)

1. Geometric closed 3-manifold groups are LERF.
2. 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 ${\Gamma=Sl_3({\mathbb Z})}$ and ${H}$ be a corner ${Sl_2({\mathbb Z})}$, answer is no. This follows from the Congruence subgroup property.

2. Proofs

Once fibering is established, the genus and rank statement follows easily. Indeed, thanks to LERF, exact sequences yield exact sequences of profinite completions. ${b_1=1}$ 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 ${\phi}$ of ${Sl_2({\mathbb Z})}$ (Jorgensen). The cardinality of the torsion part of ${H_1}$ is ${|trace(\phi)-2|}$, so this leaves only finitely many possibilities.

2.1. Proof of BRW

Assume that ${M}$ is finite volume non-compact hyperbolic. Let ${N}$ be profinite equivalent to ${M}$. Start with a few reductions.

Assume ${N}$ is compact with a single incompressible torus boundary component. Task: show that ${N}$ is fibered.

Assume that ${N}$ is finite volume hyperbolic. ${b_1=1}$ implies that ${\Delta}$ surjects onto ${{\mathbb Z}}$ with kernel ${K}$. Freedman shows that ${K}$ contains a closed surface group ${H}$. Wise shows that ${\Delta}$ is LERF, so exact sequences go to profinite completions.

Beware that if ${F}$ is a free group, ${\hat F}$ 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.