Dimension of character varieties for 33-manifolds


Let MM be a 33-manifold, compact with boundary and Γ\Gamma its fundamental group. Consider a complex reductive algebraic group G. The character variety X(Γ,G)X(\Gamma,G) is the GIT quotient Hom(Γ,G)//G\mathrm{Hom}(\Gamma,G)//G of the space of morphisms ΓG\Gamma\to G by the natural action by conjugation of GG. In the case G=SL(2,C)G=\mathrm{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)X(\Gamma,G) in terms of the Euler characteristic χ(M)\chi(M) of MM, the number tt of torus boundary components of MM, the dimension dd and the rank rr of GG. Indeed, under mild assumptions on an irreducible component X0X_0 of X(Γ,G)X(\Gamma,G), we prove the inequality dim(X0)trdχ(M).\mathrm{dim}(X_0)\geq t \cdot r - d\chi(M).Comment: 12 pages, 1 figur

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