Profinite rigidity for Seifert fibre spaces

TL;DR

This paper proves that closed orientable Seifert fibre spaces are uniquely identified by their profinite completions of fundamental groups, with some exceptions, advancing understanding of 3-manifold classification.

Contribution

It completely solves the profinite rigidity problem for closed orientable Seifert fibre spaces, identifying when they are distinguished by their profinite completions.

Findings

Most Seifert fibre spaces are distinguished by their profinite completions.

Identifies exceptions previously known due to Hempel.

Characterizes when bounded Seifert fibre space groups have isomorphic profinite completions.

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.