Some 3-manifold groups with the same finite quotients

TL;DR

This paper presents examples of 3-manifold groups that are non-isomorphic yet share identical finite quotients, highlighting limitations of profinite invariants in distinguishing these groups.

Contribution

It provides explicit examples of 3-manifold groups with identical finite quotients despite non-isomorphism, including both closed and bounded cases, especially Seifert Fibered Spaces.

Findings

Non-isomorphic 3-manifold groups can have identical finite quotients.

Examples include closed and bounded Seifert Fibered Spaces with zero rational Euler number.

Most such manifolds exhibit these properties, illustrating limitations of finite quotients as invariants.

Abstract

We give examples of closed, oriented 3-manifolds whose fundamental groups are not isomorphic, but yet have the same sets of finite quotient groups; hence the same profinite completions. We also give examples of compact, oriented 3-manifolds with non-empty boundaries whose fundamental groups though isomorphic have distinct peripheral structures, but yet have the same sets of finite peripheral pair quotients (defined below). The examples are Seifert Fibered Spaces with zero rational Euler number; moreover most of these manifolds give rise to such examples.