Poisson maps between character varieties: gluing and capping (Q6042873)

For \(\Sigma_{n,g}\) a surface of genus \(g\) with \(n\) boundary circles (or \(n\) punctures), and \(G\) either a complex reductive affine algebraic group or compact Lie group, the moduli space of representations of the fundamental group \(\pi _1(\Sigma_{n,g})\) into \(G\), the \(G\)-character variety of \(\pi _1(\Sigma_{n,g})\), has a natural Poisson structure. The authors explore induced mappings between \(G\)-character varieties of surface groups by mappings between corresponding surfaces and show that these mappings are generally Poisson. They also give an effective algorithm to compute the Poisson bi-vectors when \(G=\mathrm{SL}(2,\mathbb{C})\). They demonstrate this algorithm by explicitly calculating the Poisson bi-vector for the 5-holed sphere, the first example for an Euler characteristic-3 surface.