Faces of the scl norm ball

TL;DR

This paper explores the geometric structure of the stable commutator length norm ball in the fundamental group of a surface, revealing connections to hyperbolic geometry and quasimorphisms.

Contribution

It establishes a link between the faces of the stable commutator length ball and the rotation quasimorphism derived from hyperbolic structures on surfaces.

Findings

The projective class of the boundary chain intersects the interior of a codimension one face.

The associated homogeneous quasimorphism is the rotation quasimorphism from hyperbolic actions.

Homologically trivial 1-chains rationally cobound immersed surfaces even without boundary.

Abstract

Let F be the fundamental group of S, where S is a compact, connected, oriented surface with negative Euler characteristic and nonempty boundary. (1) The projective class of the chain \partial S in B_1(F) intersects the interior of a codimension one face of the unit ball in the stable commutator length pseudo-norm. (2) The unique homogeneous quasimorphism on F dual to this face (up to scale and elements of H^1) is the rotation quasimorphism associated to the action of F on the ideal boundary of the hyperbolic plane, coming from a hyperbolic structure on S. These facts follow from the fact that every homologically trivial 1-chain in S rationally cobounds an immersed surface with a sufficiently large multiple of the boundary. This is true even if S has no boundary.