Character Varieties

Our initial motivation was to unify the theory of spherical CR-structures and the theory of hyperbolic structures on 3-manifolds, by viewing them in the common framework of so-called flag structures. In terms of holonomy representations, which take values in \(\mathrm{SL}(2,\mathbb{C})\) and \(SU(2,1)\) for hyperbolic and spherical CR-structures respectively, we view these two groups as subgroups of \(SL(3,\mathbb{C})\).

We intend to study representations of 3-manifold groups into \(SL(3,\mathbb{C})\) that come from geometric structures on 3-manifolds. More specifically, we study representations that come from a triangulation of the 3-manifold. With a suitable interpretation, one can think of both hyperbolic and spherical CR structures as being described by associating to each vertex of the tetrahedra in the triangulation a flag, i.e. a point in the complex projective plane together with a projective line that contains that point. We shall refer to such a structure as a flag-structure on the 3- manifold. These are parametrized by the invariants of quadruples of flags. The gluing of tetrahedra in the triangulation imposes certain relations between the invariants, which can be expressed as a system of algebraic equations.

One goal is to develop efficient methods to solve these compatibility equations, and to develop a computer program that finds all solutions and computes their invariants. Eventually, we intend to study the existence and rigidity properties of the solutions, and to give a geometric interpretation of the results.

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