Right-veering diffeomorphisms of compact surfaces with boundary II

Abstract

We continue our study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary, introduced in [Invent. Math. 169 (2007) 427–449]. We conduct a detailed study of the case when the surface is a punctured torus; in particular, we exhibit the difference between the monoid of right-veering diffeomorphisms and the monoid of products of positive Dehn twists, with the help of the Rademacher function. We then generalize to the braid group $B_n$ on $n$ strands by relating the signature and the Maslov index. Finally, we discuss the symplectic fillability in the pseudo-Anosov case by comparing with the work of Roberts [Proc. London Math. Soc. (3) 82/83 (2001) 747–768/443–471].