Dimension of character varieties for 3-manifolds

Elisha Falbel (1, 2) , Antonin Guilloux (3, 1, 2)

1 UPMC - Université Pierre et Marie Curie - Paris 6
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
3 OURAGAN - OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs

Let 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).

Domaines: Topologie géométrique [math.GT]

Elisha Falbel, Antonin Guilloux. Dimension of character varieties for 3-manifolds. Proceedings of the American Mathematical Society, 2016, ⟨10.1090/proc/13394⟩. ⟨hal-01370284⟩