Coordinate rings of some \(\mathrm{SL}_2\)-character varieties (Q6589654)

Character varieties of surface groups have been widely studied, particularly regarding the computation of some algebraic invariants like their \(E\)-polynomial. They also play a prominent role in the topology of \(3\)-manifolds, starting with the foundational work of \textit{M. Culler} and \textit{P. B. Shalen} [Ann. Math. (2) 117, 109--146 (1983; Zbl 0529.57005)], where the authors used algebro-geometric properties of \(SL_2(\mathbb{C})\)-character varieties to provide new proofs of remarkable results, such as Thurston's theorem that says that the space of hyperbolic structures on an acylindrical \(3\)-manifold is compact, or the Smith conjecture in the paper mentioned above. Character varieties of \(3\)-manifolds allow even to study knots \(K \subset S^3\), by analyzing the character variety associated to the fundamental group of their complement, \(\Gamma_K =\pi_1(S^3-K)\). \par The authors determine generators of the coordinate ring of \(SL_2\)-character varieties. In the case of the free group \(F_3\) they obtain an explicit equation of the \(SL_2\)-character variety. For free groups \(F_k\), the authors find transcendental generators. Finally, for the case of the \(2\)-torus, an explicit equation of the \(SL_2\)-character variety is presented and then used the description to compute their \(E\)-polynomials.