The infinite direct product of Dehn twists acting on infinite dimensional Teichmüller spaces
The author considers the action of the modular group Mod\((S)\) on the (infinite-dimensional) Teichmüller space \(T(S)\) of a topologically infinite Riemann surface \(S\). It is known that orbits of points under this action are not always discrete (in contrast to the case of topologically finite surfaces). In a previous paper, the author had studied orbits of points under a cyclic subgroup of Mod\((S)\) generated by an elliptic element of infinite order. In the paper under review, he considers orbits under a parabolic abelian subgroup \(G\) generated by an infinite number of Dehn twists along mutually disjoint simple closed curves. (Here, the definition of elliptic and parabolic are the same as those introduced by Bers in the case of finite-dimensional Teichmüller spaces, that is, an element \(g\) in Mod\((S)\) is elliptic if it has a fixed point in \(T(S)\) and it is parabolic if we have \(\inf_{p\in T(S)}d(p,g(p))=0\), \(d\) being the Teichmüller distance the infimum being taken over all points \(p\) in \(T(S)\).) The author proves that convergence of such an orbit with respect to the Teichmüller distance implies locally uniform convergence of the elements of the modular group, which implies in particular that the orbit is always closed. By estimating the maximal dilatation of an element of \(G\) in terms of hyperbolic lengths of simple closed geodesics, he obtains necessary and sufficient conditions for the orbit to be discrete and countable.
scientific article; zbMATH DE number 2091926