On analytic properties of deformation spaces of Kleinian groups (Q2790741)

Let \(G_0\) be a finitely generated non-elementary Kleinian group, that is, a finitely generated discrete subgroup \(G_0\) of \(\text{PSL}(2,\mathbb{C})\) in which there are two elements \(A\), \(B\) of infinite order with \(\text{tr\,}ABA^{-1}B^{-1}\neq 2\). The character space of \(G_0\) is the space \(\widetilde{\Hom}(G_0, \text{PSL}(2,\mathbb{C}))\) of homomorphism from \(G_0\) to \(\text{PSL}(2, \mathbb{C})\) modulo conjugation in \(\text{PSL}(2,\mathbb{C})\).NEWLINENEWLINE The deformation space \(D(G_0)\) of \(G_0\) consists of those \(\theta\in\widetilde{\Hom}(G_0, \text{PSL}(2,\mathbb{C}))\) for which there exists a quasiconformal map \(\omega:\widehat G\to\widehat{\mathbb{C}}\) fixing \(0\), \(1\) and \(\infty\) such that \(\theta\) is represented by an isomorphism \(\varphi:G_0\to \varphi(G_0)\subset\text{PSL}(2,\mathbb{C})\), \(g\mapsto\omega g\omega^{-1}\). \(D(G_0)\) is a complex manifold, and it is holomorphically converse.NEWLINENEWLINE In this paper, the author considers the complex properties of \(D(G_0)\) and describes certain analytic structures of \(D(G_0)\), some of which give improvements of results by Bers, Kra, Maskit and McMullen.