Dimension of character varieties for $3$-manifolds

TL;DR

This paper establishes a lower bound on the dimension of irreducible components of character varieties for 3-manifolds, linking topological features with algebraic properties of the representation space.

Contribution

It generalizes Thurston's work by providing a dimension bound for character varieties of 3-manifolds with boundary for arbitrary complex reductive groups.

Findings

Derived a lower bound for the dimension of character variety components

Connected topological invariants with algebraic properties of representation spaces

Extended Thurston's results beyond SL(2,C) to general reductive groups

Abstract

Let $M$ be a $3$-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, as presented by Culler-Shalen, 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 $$\mathrm{dim}(X_0)\geq t \cdot r - d\chi(M).$$