Some Pseudo-anosov Maps on Punctured Riemann Surfaces Represented by Multi-twists

Let A, B be families of disjoint non-trivial simple closed geodesics on a Riemann surface S so that each component of S\{A∪ B} is either a disk or a once punctured disk. Let w be any word consisting of Dehn twists along elements of A and inverses of Dehn twists along elements of B so that the Dehn twist along each element of A and the inverse of the Dehn twist along each element of B occur at least once in w. It is well known that w represents a pseudoAnosov class. In this paper we study those pseudo-Anosov maps f on S projecting to the trivial map as a puncture a is filled in. We prove the following theorem. Let S be of type (p, n), 3p− 4 + n > 0 and n ≥ 1. Write A = {α1, . . . , αk}, k ≥ 1. If all αi are non-trivial and distinct as geodesics on S̃ = S ∪ {a}, then for any integer tuples (n1, . . . , nk), the composition t n1 α1 ◦· · ·◦tk αk ◦f is also a pseudo-Anosov map. As a consequence we also prove that if S is of type (p, 1) with p ≥ 2, then for every integer m, f is not isotopic to any word w described above.