Distinguishing geometries using finite quotients

Abstract

We prove that the profinite completion of the fundamental group of a compact 3-manifold M satisfies a Tits alternative: if a closed subgroup H does not contain a free pro-p subgroup for any p, then H is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of the fundamental group of a closed, hyperbolic 3-manifold does not contain a subgroup isomorphic to Z 2 . This gives a profinite characterization of hyperbolicity among irreducible 3-manifolds. We also characterize Seifert fibred 3-manifolds as precisely those for which the profinite completion of the fundamental group has a non-trivial procyclic normal subgroup. Our techniques also apply to hyperbolic, virtually special groups, in the sense of Haglund and Wise. Finally, we prove that every finitely generated pro-p subgroup of the profinite completion of a torsion-free, hyperbolic, virtually special group is free pro-p.In a heuristic commonly used to distinguish 3-manifolds M and N, one computes all covers M 1 , . . . , M m of M and N 1 , . . . , N n of N up to some small degree, and then compares the resulting finite lists (M i ) and (N j ). If they can be distinguished, then one has a proof that M and N were not homeomorphic.It is natural to ask whether this method always works. A more precise question was formulated by Calegari-Freedman-Walker in [CFW10], who asked whether the fundamental group of a 3-manifold is determined by its finite quotients, or equivalently, by its profinite completion. (A standard argument shows that two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. In fact, the Geometrization theorem is equivalent to the Hyperbolization theorem and the Seifert conjecture, together with the Elliptization theorem, which asserts that M is spherical if and only if π 1 M is finite [Sco83]. Since 3-manifold groups are residually finite [Hem87], π 1 M is finite if and only if its profinite completion is, so there is no distinct profinite analogue of the Elliptization theorem. Theorems A and B can therefore be thought of as providing 2 a complete profinite analogue of the Geometrization theorem (although one should note, of course, that our proofs rely essentially on Geometrization). In Theorem 8.4, we proceed to show that the profinite completion of the fundamental group detects the geometry of a closed, orientable, irreducible 3-manifold.Alternatively, Theorem A can be thought of as a classification result for the abelian subgroups of profinite completions of fundamental groups of hyperbolic manifolds. In fact, we prove a much more general 'Tits alternative' for profinite completions of 3-manifold groups. We use the notation Z π to denote p∈π Z p .Theorem C. If M is any compact 3-manifold and H is a closed subgroup of π 1 M that does not contain a free non-abelian pro-p subgroup for any prime p then H is on the following list:1. H is conjugate to the completion of a virtually soluble subgroup of π 1 M; or, where π and σ a...

Pro-p subgroups of profinite completions of 3-manifold groups

“…This was stated in [, p. 376], and a careful proof was written down in [, Proposition 6.8]. Proposition follows in a straightforward manner from the following statement.…”

“…Proof First note that a non‐trivial stabilizer of any edge coincides with one of its vertex stabilizers or , say , since otherwise by [, Lemma 11.2] the stabilizers of two vertices of do not generate a pro‐ group. It follows that a maximal connected subgraph of having non‐trivial stabilizers in of all its edges has diameter at most .…”

“…Proposition and Lemma enable us to generalize [, Theorem 11.1] from the 1‐acylindrical case to the k ‐acylindrical case, for any natural number . Theorem Let be a finitely generated pro‐ group acting ‐acylindrically on a profinite tree .…”

“…Thus collapsing all but one edges of this geodesic and their translates, we may assume that has just one edge. Now we can apply [, Lemma 11.2 ] to deduce that is not pro‐ as required.…”

“…The work of this paper was partially motivated by [, Theorem F], where it was proved that if is hyperbolic then any finitely generated pro‐ subgroup of is free. In fact the proof of Theorem follows the same strategy as in . By applying first the (profinite) Kneser–Milnor decomposition and then the (profinite) Jaco–Shalen–Johannson (JSJ) decomposition, the theorem is reduced to the cases of Seifert‐fibred and hyperbolic manifolds, possibly with cusps.…”