Dimension of character varieties for 3-manifolds

2015/10/02 by Falbel, Elisha, Guilloux, Antonin
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1510.00567

Abstract

Let M be a 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,\mathbb C) this space has been thoroughly studied. Following work of Thurston, as presented by Culler-Shalen, 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).

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