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.
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Acknowledgements
We would like to thank L. Charles, D. Bertrand and P. Philippon for their kind interest and P. Habegger for guiding us through his work on effective bounded height properties. Also, we would like to thank the referee for all his comments, and for suggesting to us the application of Masser’s theorem \({\mathbb {G}}_{\mathrm {m}}\) and Lemma 5.2 of [1], which led to Theorem 3.6 of the present article. A different question related to character varieties of Dehn filling was solved with the same kind of tools by B. Jeon in [15]. We thank I. Agol for pointing it to us. More recently, similar results about “singular intersections” of Zilber–Pink type have been proved in a different setting with different approaches by several authors, namely for sections of elliptic surfaces \(E\rightarrow C\) (see [5, 22]).
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Marché, J., Maurin, G. Singular intersections of subgroups and character varieties. Math. Ann. 386, 713–734 (2023). https://doi.org/10.1007/s00208-022-02414-8
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DOI: https://doi.org/10.1007/s00208-022-02414-8