A lower bound for dilatations of certain class of pseudo-Anosov maps of Riemann surfaces

Let S be a Riemann surface of type (p, n) with 3p + n > 4 that contains at least one puncture a. Let p,n denote the set of pseudo-Anosov maps of S that are isotopic to products of two Dehn twists and are isotopic to the identity map on ˜ S = S ∪ {a}. In this article, we give a lower bound for dilatations of elements of p,n. We also estimate for any hyperbolic structure of ˜ S the hyperbolic lengths of those filling closed geodesics of ˜ S stemming from the elements of p,n