On the action of the mapping class group for Riemann surfaces of infinite type (Q1768153)

The mapping class group Mod\( (R)\) for a Riemann surface \(R\) is the set of equivalence classes of quasiconformal automorphisms of \(R.\) Two quasiconformal automorphisms \(h_{1}\) and \(h_{2}\) of \(R\) are equivalent if \(h_{2}^{-1}\circ h_{1}\) is homotopic to the identity by a homotopy that keeps every point of ideal boundary \(\partial R\) fixed throughout. This is a group of the biholomorphic automorphisms of the Teichmueller space and it acts faithfully and properly discontinuously. We say that a subgroup \(G\) of Mod\((R)\) is discrete if the orbit of any point of \(T(R)\) under the \(G\) action is discrete. The reduced mapping class group Mod\(^{\sharp}(R)\) is the set of homotopy classes of quasiconformal automorphisms of \(R\) whose homotopy maps do not necessarily keep points of \(\partial R\) fixed. Theorem 1. Let \(R\) be a Riemann surface with the non-abelian fundamental group. Suppose that \(R\) satisfies the following two conditions: (1) There exists a constant \(\varepsilon > 0\) such that the \(\varepsilon\)-thin part of \(R\) consists only of cusp neighborhoods. (2) There exists a constant \(M > 0\) and a connected component \(R^{\ast}_{M}\) of \(R_{M}\) such that the homomorphism of \(\pi_{1}(R^{\ast}_{M})\) to \(\pi_{1}(R)\) which is induced by the inclusion map of \(R^{\ast}_{M}\) to \(R\) is surjective. Then Mod\(_{c}^{\sharp}(R)\) is discrete for any simple closed geodesic \(c\) on \(R.\) Theorem 2. Let \(R\) be a Riemann surface satisfying the conditions in Theorem 1. Suppose that either the genus, the number of cusps or the number of holes of \(R\) is positive finite. Then Mod\(^{\sharp}(R)\) is discrete.