The fractional Dehn twist coefficient (FDTC) is a rational number associated to a mapping class on a (finite-type) surface with boundary. This 2-dimensional invariant has many applications to 3-manifold topology and contact geometry. One way to think of the FDTC is as a real-valued function on the mapping class group of a surface with many nice properties. In this talk, we will give sufficient conditions on a more general group to admit a function which behaves like the FDTC. In particular, we use this to generalize the FDTC to infinite-type surfaces (with boundary); in this setting, we show that the "fractional" Dehn twist coefficient need not be rational. This is joint work with Peter Feller and Diana Hubbard.