Triangle groups, automorphic forms, and torus knots

Abstract

This article is concerned with the relation between several classical and well-known objects: triangle Fuchsian groups, ${\mathbb{C}}^{\times }$-equivariant singularities of plane curves, torus knot complements in the 3-sphere. The prototypical example is the modular group $PS{L}_2(\mathbb{Z})$: the quotient of the nonzero tangent bundle on the upper-half plane by the action of $PS{L}_2(\mathbb{Z})$ is biholomorphic to the complement of the plane curve $z^3-27w^2=0$. This can be shown using the fact that the algebra of modular forms is doubly generated, by $g_2,g_3$, and the cusp form $\Delta =g_2^3-27g_3^2$ does not vanish on the half-plane. As a byproduct, one finds a diffeomorphism between $PS{L}_2(\mathbb{R})/PS{L}_2(\mathbb{Z})$ and the complement of the trefoil knot - the local knot of the singular curve. This construction is generalized to include all $(p,q,\infty )$-triangle groups and, respectively, curves of the form $z^q+w^p=0$ and $(p,q)$-torus knots, for $p,q$ co-prime. The general case requires the use of automorphic forms on the simply connected group $\widetilde{S{L}_2}(\mathbb{R})$. The proof uses ideas of Milnor and Dolgachev, which they introduced in their studies of the spectra of the algebras of automorphic forms of cocompact triangle groups (and, more generally, uniform lattices). It turns out that the same approach, with some modifications, allows one to handle the cuspidal case.