Triangle groups, automorphic forms, and torus knots

Abstract

This paper's theme is the relation between several classical and well-known objects: triangle Fuchsian groups, quasi-homogeneous singularities of plane curves, torus knot complements in the 3-sphere. Torus knots are the only nontrivial knots whose complements admit transitive Lie group actions. In fact S^3\K_{p,q} is diffeomorphic to a coset space of the universal covering group of PSL_2(R) with respect to a discrete subgroup G contained in the preimage of a (p,q,\infty)-triangle Fuchsian group. The existence of such a diffeomorphism between is known from a general topological classification of Seifert fibred 3-manifolds. Our goal is to construct an explicit diffeomorphism using automorphic forms. Such a construction is previously known for the trefoil knot K_{2,3} and in fact S^3\K_{2,3} = SL_2(R)/SL_2(Z). The connection between the two sides of the diffeomorphism comes via singularities of plane curves.