Combable functions, quasimorphisms, and the central limit theorem
by
Danny Calegari
Caltech
Coauthors: Koji Fujiwara

We discuss a circle of ideas involving interactions between bounded cohomology, ergodic theory, and computer science. These seemingly disparate tools are unified in the context of hyperbolic groups by the theme of locality. We introduce the class of combable and bicombable functions, roughly, functions whose discrete derivatives can be calculated by certain simple machines. We show that the Epstein-Fujiwara quasimorphisms are bicombable; conversely, any bicombable antisymmetric function on a hyperbolic group is a quasimorphism, thus showing that quasimorphisms arise naturally in the context of automatic group theory. The finite structure of a bicombable function leads to a simple structure theory for the distribution of its values on a group. This lets us prove that the distribution of values of the Epstein-Fujiwara quasimorphisms on hyperbolic groups satisfy a central limit theorem. This is joint work with Koji Fujiwara.

Date received: January 24, 2008