The action of geometric automorphisms of asymptotic Teichmüller spaces (Q874427)

Let \(R\) be a Riemann surface and let \(T(R)\) be the corresponding Teichmüller space. Then the Teichmüller modular group \(\mathrm{Mod}(R)\) operates on \(T(R)\). If \(R\) is of finite type the operation is discontinuous. This is no longer true if \(R\) is of infinite analytic type. To study this phenomenon Gardiner and Sullivan introduced the notion of the asymptotic Teichmüller space. This is a quotient space of the Teichmüller space and the geometric automorphism group \(\mathrm{G}(R)\), an image of the mapping class group, acts on the asymptotic Teichmüller space. The action of this group can be trivial; the author proves that if \(R\) satisfies the upper bound condition then the action is non-trivial. The author also studies the limit sets of the action of \(\mathrm{G}(R)\) and shows in particular that, even if \(R\) satisfies both the upper and lower bound conditions then this limit set can be non-trivial while that of the Teichmüller modular group is empty.