Singular intersections of subgroups and character varieties

Abstract

We prove a global local rigidity result for character varieties of 3-manifolds into \(\mathrm {SL}_2({\mathbb {C}})\). Given a 3-manifold with toric boundary M satisfying some technical hypotheses, we prove that all but a finite number of its Dehn fillings \(M_{p/q}\) are globally locally rigid in the following sense: every irreducible representation \(\rho :\pi _1(M_{p/q})\rightarrow \mathrm {SL}_2({\mathbb {C}})\) is infinitesimally rigid, meaning that \(H^1(M_{p/q},\text {Ad}_\rho )=0\). This question arose from the study of asymptotics problems in topological quantum field theory developed in Charles and Marché (Knot state asymptotics II. Irreducible representations and the Witten conjecture, 2011). The proof relies heavily on recent progress in diophantine geometry and raises new questions of Zilber–Pink type. The main step is to show that a generic curve lying in a plane multiplicative torus intersects transversally almost all subtori of codimension 1. We prove an effective result of this form, based mainly on a height upper bound of Habegger.