Taut polynomials from finite quotients of fibered hyperbolic 3-manifold groups

Abstract

We prove that, whenever the monodromy maps are fully-punctured, the finite quotients of a fibered hyperbolic $3$-manifold group detect the taut polynomials of the fibered faces of the Thurston norm ball, as well as the Teichmüller polynomials of suitable Dehn fillings. Toward this, we develop a general framework for the profinite invariance of twisted multivariable Alexander polynomials, which is of independent interest; we also prove that being fully-punctured and preserving transverse orientations are profinite properties of the monodromy maps. As an application, we identify specific one-cusped hyperbolic $3$-manifolds \blue{whose fundamental groups} are profinitely rigid among $3$-manifold groups, by a strategy using normalized dilatations and the veering census; notably, the taut polynomials distinguish a pair of mapping tori sharing the fiber and the dilatation.

37 pages, 2 figures. v2 contains new contents on transverse orientations (Subsec.1.4 and Sec.6) and major updates on algebraic theory of multivariable polynomials (Subsec.3.3--3.5)