Dimension of character varieties for 3-manifolds

International audienceLet M be an orientable 3-manifold, compact with boundary and Γ its fundamental group. Consider a complex reductive algebraic group G. The character variety X(Γ, G) is the GIT quotient Hom(Γ, G)//G of the space of morphisms Γ → G by the natural action by conjugation of G. In the case G = SL(2, C) this space has been thoroughly studied. Following work of Thurston [Thu80], as presented by Culler-Shalen [CS83], we give a lower bound for the dimension of irreducible components of X(Γ, G) in terms of the Euler characteristic χ(M) of M , the number t of torus boundary components of M , the dimension d and the rank r of G. Indeed, under mild assumptions on an irreducible component X0 of X(Γ, G), we prove the inequality dim(X0) ≥ t · r − dχ(M)