Entropy, Weil-petersson Translation Distance and Gromov Norm for Surface Automorphisms

Thanks to a theorem of Brock on the comparison of Weil-Petersson translation distances and hyperbolic volumes of mapping tori for pseudoAnosovs, we prove that the entropy of a surface automorphism in general has linear bounds in terms of a Gromov norm of its mapping torus from below and an inbounded geometry case from above. We also prove that the WeilPetersson translation distance does the same from both sides in general. The proofs are in fact immediately derived from the theorem of Brock, together with some other strong theorems and small observations.