Character Varieties

TL;DR

This paper investigates the structure and properties of character varieties associated with finitely generated groups and reductive algebraic groups, including their tangent spaces, symplectic structures, and Lagrangian submanifolds in the context of 3-manifolds.

Contribution

It generalizes the tangent space formula for character varieties and proves that certain subsets form isotropic or Lagrangian submanifolds in the setting of 3-manifold boundary representations.

Findings

Tangent spaces described via first cohomology groups.

Subset of representations extending to the manifold forms an isotropic submanifold.

Reduced points of the character variety correspond to Lagrangian submanifolds.

Abstract

We study properties of irreducible and completely reducible representations of finitely generated groups Gamma into reductive algebraic groups G in in the context of the geometric invariant theory of the G-action on Hom(Gamma,G) by conjugation. In particular, we study properties of character varieties, X_G(Gamma)=Hom(Gamma,G)//G. We describe the tangent spaces to X_G(Gamma) in terms of first cohomology groups of Gamma with twisted coefficients, generalizing the well known formula. Let M be an orientable 3-manifold with a connected boundary F of genus > 1 and let X_G^g(F) be the subset of the G -character variety of F composed of conjugacy classes of good representations. By a theorem of Goldman, X_G^g(F) is a holomorphic symplectic manifold. We prove that the set of good G-representations of pi_1(F) which extend to representations of pi_1(M) is an isotropic submanifold of X_G^g(F). If these representations correspond to reduced points of the G-character variety of M then this submanifold is Lagrangian.