Decomposition of integrable holomorphic quadratic differential on Riemann surface of infinite type

In this paper, decomposition of the integrable holomorphic quadratic differential on Riemann surface of infinite type is studied. Hubbard, Schleicher and Shishikura gave a thick-thin decomposition on Riemann surface of finite type with an integrable holomorphic quadratic differential and they thought their result might be generalized to arbitrary hyperbolic Riemann surface of infinite type. We confirm what they thought is right and give a proposition (Proposition 2.2) of its own interest.