Homeomorphisms and intersection numbers (Q1657940)

Let \(S\) be a closed, orientable, hyperbolic surface. A (not necessarily orientation-preserving) mapping class induces an intersection number-preserving map on the space of measured lamination \({\mathcal ML}(S)\) and its projectivization \({\mathcal PML}(S)\), where in the latter case only triviality and nontriviality of the intersection number are well-defined. It is known from \textit{N. V. Ivanov} [Int. Math. Res. Not. 1997, No. 14, 651--666 (1997; Zbl 0890.57018)] that, except for the action of the hyperelliptic involution of a genus 2 surface, any nontrivial mapping class acts nontrivially on the curve complex and hence on \({\mathcal ML}(S)\) and its projectivization. The paper under review proves that any intersection-number preserving automorphism of \({\mathcal ML}(S)\) is induced by a mapping class. Hence the mapping class group is isomorphic to \(Aut({\mathcal ML}(S))\) for genus \(g\geq 3\), while for genus 2 one has an epimorphism with kernel of order 2. Moreover the authors prove that an automorphism of \({\mathcal PML}(S)\) is induced by a mapping class if and only if it preserves triviality and nontriviality of the intersection number.