The Dehn twist coefficient for big and small mapping class groups

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 . In contrast to this, we find that for some surfaces of infinite type the DTC has image all of . 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.