Profinite rigidity and hyperbolic four-punctured sphere bundles over the circle

TL;DR

This paper proves that hyperbolic four-punctured sphere bundles over the circle are uniquely identified by the finite quotients of their fundamental groups, advancing the understanding of profinite rigidity in 3-manifold topology.

Contribution

It extends Liu's result to show the topological type of fibers in hyperbolic 3-manifolds is determined by their profinite completions, specifically for four-punctured sphere bundles.

Findings

Hyperbolic four-punctured sphere bundles are distinguished by their fundamental group quotients.

Profinite completion detects the topological fiber type in hyperbolic fibered 3-manifolds.

The result enhances the understanding of profinite rigidity in 3-manifold groups.

Abstract

We show that hyperbolic four-punctured $S^2-$bundles over $S^1$ are distinguished by the finite quotients of their fundamental groups among all 3-manifold groups. To do this, we upgrade a result of Liu to show that the topological type of a fiber is detected by the profinite completion of the fundamental group of a fibered hyperbolic 3-manifold.