Dimension of character varieties for $3$-manifolds

Abstract:

Let $M$ be an orientable $3$-manifold, compact with boundary and $\Gamma$ its fundamental group. Consider a complex reductive algebraic group $G$. The character variety $X(\Gamma ,G)$ is the GIT quotient $\mathrm {Hom}(\Gamma ,G)//G$ of the space of morphisms $\Gamma \to G$ by the natural action by conjugation of $G$. In the case $G=\mathrm {SL}(2,\mathbb {C})$ this space has been thoroughly studied.

Following work of Thurston (1980), as presented by Culler-Shalen (1983), we give a lower bound for the dimension of irreducible components of $X(\Gamma ,G)$ in terms of the Euler characteristic $\chi (M)$ 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 $X_0$ of $X(\Gamma ,G)$, we prove the inequality \[ \mathrm {dim}(X_0)\geq t \cdot r - d\chi (M).\]