The invariant differential forms on the Teichmüller space under the Fenchel-Nielsen flows (Q1203591)

Let \(T_ g\) resp. \(M_ g\) be the Teichmüller space resp. moduli space of compact Riemann surfaces of genus \(g\). To each simple closed curve on the closed surface \(\Sigma_ g\) of genus \(g\) is associated the Fenchel- Nielsen flow on \(T_ g\) obtained by cutting the surface along the curve, rotating one boundary by an arbitrary real-valued angle and reglueing in the new position, thus creating a new Riemann surface and point in Teichmüller space. Let FN be the subgroup of diffeomorphisms of \(T_ g\) generated by the Fenchel-Nielson flows associated to all simple closed geodesics on the surface. Then FN operates transitively on \(T_ g\) and contains the Teichmüller modular group or mapping class group \({\mathfrak M}_ g\) of \(\Sigma_ g\) as a ``discrete'' subgroup (generated by Dehn twists around simple closed curves, that is rotations by \(2\pi)\), with \(T_ g/{\mathfrak M}_ g=M_ g\). The main result of the paper computes the differential forms \(\Omega^*(T_ g)^{FN}\) on \(T_ g\) invariant under the action of FN; more precisely in dimensions smaller or equal to \(3g-5\) one has \(\Omega^*(T_ g)^{FN}=\mathbb{R}[\omega]\) where \(\omega\in\Omega^ 2(T_ g)\) denotes the Weil-Petersson Kähler form. Combining this with various known results one obtains that the homomorphism \(H^*(\Omega^*(T_ g)^{FN})\to H^*(M_ g)\) induced in cohomology by the inclusion \(\Omega^*(T_ g)^{FN}\subset\Omega^*(T_ g)^{{\mathcal M}g}=\Omega^*(M_ g)\) is stably injective, stably isomorphic in dimensions 0,1 and 2, but not surjective (not even stably) in dimensions larger than 3. So the group FN does not determine the stable real cohomology of the moduli space of compact Riemann surfaces. In fact, the setting was modelled after the case of a discrete subgroup in a semi-simple Lie group where an analogous approach has been used to determine the stable real cohomology of arithmetic groups.