Profinite rigidity of fibring

TL;DR

This paper introduces TAP groups, showing algebraic fibring can be detected via twisted Alexander polynomials and proving it is a profinite property for certain classes of groups, with applications to 3-manifold groups.

Contribution

It defines TAP groups and demonstrates algebraic fibring as a profinite property for finitely presented LERF groups, extending to products of limit groups and applications in 3-manifold topology.

Findings

Finitely presented LERF groups lie in TAP_1(R) for all integral domains R.

Algebraic fibring is a profinite property for these groups.

Results apply to profinite rigidity of 3-dimensional Poincaré duality groups and RFRS groups.

Abstract

We introduce the classes of TAP groups, in which various types of algebraic fibring are detected by the non-vanishing of twisted Alexander polynomials. We show that finitely presented LERF groups lie in the class $\mathsf{TAP}_1(R)$ for every integral domain $R$, and deduce that algebraic fibring is a profinite property for such groups. We offer stronger results for algebraic fibring of products of limit groups, as well as applications to profinite rigidity of Poincar\'e duality groups in dimension $3$ and RFRS groups.