Quasiconformal mapping class groups having common fixed points on the asymptotic Teichmüller spaces

For an analytically infinite Riemann surface R, we consider the action of the quasiconformal mapping class group MCG(R) on the Teichmüller space T(R), which preserves the fibers of the projection α: T(R) → AT(R) onto the asymptotic Teichmüller space AT(R). We prove that if MCG(R) has a common fixed point α(p) AT(R), then it acts discontinuously on the fiber T _p over α(p), which is a separable subspace of T(R). In particular, this implies that MCG(R) is a countable group. This is a generalization of a fact that MCG(R) acts discontinuously on T _o = T(R) for an analytically finite Riemann surface R.