The Dehn twist coefficient for big and small mapping class groups

TL;DR

This paper introduces the Dehn twist coefficient (DTC), a new quasimorphism extending the fractional Dehn twist coefficient to infinite-type surfaces, revealing new irrational rotation behaviors and characterizing the DTC uniquely.

Contribution

It defines and characterizes the DTC as a unique homogeneous quasimorphism for infinite-type surfaces, generalizing the FDTC and uncovering irrational rotation phenomena.

Findings

DTC coincides with FDTC for finite-type surfaces

DTC can have image all of 5 for some infinite-type surfaces

Constructs maps with irrational rotation behavior on infinite-type surfaces

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 $\mathbb{Q}$. In contrast to this, we find that for some surfaces of infinite type the DTC has image all of $\mathbb{R}$. 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.