Exotic projective structures and quasi-Fuchsian space.
Let \(S\) be an orientable closed surface of genus \(g \geq 2\). Let \(P(S)\) be the space of all (marked) projective structures on \(S\). If \(V(S)=\text{Hom}(\pi_{1}(S),PSL(2,{\mathbb C}))/PSL(2,{\mathbb C})\), then we have a natural map \(hol:P(S) \to V(S)\). Inside the space \(V(S)\) is \(QF(S)\) whose elements are those (classes of) representations \(\rho\) for which \(\rho(\pi_{1}(S))\) is a quasifuchsian group. The author consider the subset \(Q(S)=hol^{-1}(QF(S))\) which turns out to be a collection of components \({\mathcal D}_{\lambda}\), each one biholomorphically equivalent to \(QF(S)\), parameterized by the set of integral points of measured laminations on \(S\). The component \({\mathcal D}_{0}\) consists of the standard projective structures and the others to the exotic ones. It is proved that \(\overline{{\mathcal D}_{0}} \cap \overline{{\mathcal D}_{\lambda}} \neq \emptyset\). It follows that in each component \({\mathcal D}_{\lambda}\) there is a sequence of exotic projective structures on \(S\) converging to a point in the relative boundary of the standard projective structures.
scientific article; zbMATH DE number 1820760
scientific article