Profinite rigidity for Seifert fibre spaces

Abstract

An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this question completely for closed orientable Seifert fibre spaces. In particular, all Seifert fibre spaces are distinguished from each other by their profinite completions apart from some previously-known examples due to Hempel. We also characterize when bounded Seifert fibre space groups have isomorphic profinite completions, given some conditions on the boundary.

Profinite detection of 3-manifold decompositions

“…In light of Theorem B, the next step in addressing Question 0.1 is to consider the pieces of the JSJ decomposition. The Seifert fibred case has been resolved by Wilkes [Wil17b], building on work of Hempel [Hem14]: Seifert fibred -manifolds are not profinitely rigid, but do have finite genus, and Wilkes was able to give a complete description of when two such -manifold groups have isomorphic profinite completions; he was subsequently able to extend this to a complete answer to Question 0.1 for graph manifolds [Wil18a, Theorem 10.9]. In that paper, Sol-manifolds were not included in the class of graph manifolds.…”

Profinite rigidity of graph manifolds, II: knots and mapping classes

“…There is a canonical injection Aut(π 1 S) ֒→ Aut( π 1 S) and we will abuse notation by identifying an automorphism of φ 1 S with the induced automorphism of the profinite completion. As was noted by Boileau and Friedl [BF15b, Corollary 3.6], Theorem 5.2 of [Wil17] implies that the canonical map Out(π 1 S) → Out( π 1 S)…”

“…There is a corresponding notion for non-orientable orbifolds, but this will not be relevant to this paper. [Wil17]). Let M and N be Seifert fibre spaces whose boundary components are ∂M 1 , .…”

“…It follows that λ • and µ • are constant on connected components of X {r}. Now on the root pieces of M and N , the fibre subgroup is the unique maximal central subgroup of the relevant profinite fundamental group by Theorem 5.5 of [Wil17] and so is preserved by Φ r . Let the map on fibre subgroups be multiplication by λ r (as usual, identifying this fibre subgroup with Z via a generator in the discrete fundamental group).…”

“…There has been a recent slew of papers dealing with the properties of 3-manifold groups which may be detected via the finite quotients of the group, or equivalently via its profinite completion. Examples of major advances in this field include the detection of fibring for compact 3-manifolds by Jaikin-Zapirain [JZ17]; the detection of the geometry of a closed 3-manifold by Wilton and Zalesskii [WZ10]; the proof that each once-punctured torus bundle is profinitely rigid [BRW17] (that is, it may be distinguished from all other 3-manifolds by the profinite completion of its fundamental group); and the classification of those Seifert fibre spaces with the same profinite completions of fundamental groups by the author [Wil17].…”