The Dehn twist coefficient for big and small mapping class groups

Abstract

We study a quasimorphism, which we call the Dehn twist coefficient (DTC), from the mapping class group of a surface (with a chosen compact boundary component) that generalizes the well-studied fractional Dehn twist coefficient (FDTC) to surfaces of infinite type. Indeed, for surfaces of finite type, the DTC coincides with the FDTC. We provide a characterization of the DTC as the unique homogeneous quasimorphism satisfying certain positivity conditions. This characterization is new even for the classical finite-type case and requires minimal input beyond elementary topology. The FDTC has image contained in $Q$. In contrast to this, we find that for some surfaces of infinite type the DTC has image all of $R$. To see this, we provide a new construction of maps with irrational rotation behavior for some surfaces of infinite type with a countable space of ends or even just one end. In fact, we find that the DTC is the right tool to detect irrational rotation behavior, even for surfaces without boundary.