The boundary of a fibered face of the magic 3-manifold and the asymptotic behavior of minimal pseudo-Anosov dilatations

Let delta(g,n) be the minimal dilatation of pseudo-Anosovs defined on an orientable surface of genus g with n punctures. It is proved by Tsai that for any fixed g >= 2, there exists a constant c(g) depending on g such that [GRAPHIC] This means that the logarithm of the minimal dilatation log delta(g,n) is on the order of log n/n. We prove that if 2g + 1 is relatively prime to s or s + 1 for each 0 <= s <= g, then [GRAPHIC] holds. In particular, if 2g + 1 is prime, then the above inequality on delta(g,n) holds. Our examples of pseudo-Anosovs phi's which provide the upper bound above have the following property: The mapping torus M-phi of phi is a single hyperbolic 3- manifold N called the magic manifold, or the fibration of M-phi comes from a fibration of N by Dehn filling cusps along the boundary slopes of a fiber.