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

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.

arXiv comment11 pages, 5 figures. Significant corrections to Version 2 caused by an error in the attempted proof of Theorems 1.1 - 1.3 (Version 2). The critical mistake (in Lemma 5.1 (Version 2)) was not distinguishing between pseudoAnosov elements of mapping class groups and conjugacy classes of pseudoAnosovs