Extremality of the rotation quasimorphism on the modular group

Abstract

In this talk, I will discuss the notions of stable commutator length and homogeneous quasimorphism, which are related by a duality theorem of Bavard. After introducing these notions, I will restrict attention to a particular group, the modular group PSL(2,Z), and a particular quasimorphism, the rotation quasimorphism rot. We study for which elements A of PSL(2,Z) Bavard's bounds are sharp in the sense that scl(A) = rot(A)/12. First I will describe some experimental results based on computation of stable commutator length. Then I will discuss the following stability theorem: for any element of the modular group, the product of this element with a sufficiently large power of a parabolic element is an element that satisfies scl = rot/12. This result is joint work with Danny Calegari.