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

TL;DR

This paper explores how the profinite fundamental groups can distinguish graph manifold exteriors and analyzes the behavior of knots and mapping classes, revealing new insights into 3-manifold group classifications.

Contribution

It demonstrates that graph knot exteriors are uniquely identified by their profinite fundamental groups and establishes a strong conjugacy separability result for specific surface mapping classes.

Findings

Graph knot exteriors are distinguished by their profinite fundamental groups.

Proved strong conjugacy separability for certain surface mapping classes.

Enhanced understanding of 3-manifold group classification through profinite invariants.

Abstract

In this paper we study some consequences of the author's classification of graph manifolds by their profinite fundamental groups. In particular we study commensurability, the behaviour of knots, and relation to mapping classes. We prove that the exteriors of graph knots are distinguished among all 3-manifold groups by their profinite fundamental groups. We also prove a strong conjugacy separability result for certain mapping classes of surfaces.