Faces of the scl norm ball

Abstract

Let F = π 1 (S) where S is a compact, connected, oriented surface with χ(S) < 0 and nonempty boundary.(1) The projective class of the the chain ∂S ∈ B H 1 (F ) intersects the interior of a codimension one face π S of the unit ball in the stable commutator length norm on B H 1 (F ). ( 2) The unique homogeneous quasimorphism on F dual to π S (up to scale and elements of H 1 (F )) is the rotation quasimorphism associated to the action of π 1 (S) 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 C in S rationally cobounds an immersed surface with a sufficiently large multiple of ∂S. This is true even if S has no boundary.

Stable commutator length is rational in free groups

“…Secondly, the polyhedral structure on the scl norm unit ball is studied in [6], and it is shown that every realization of a free group as 1 ( ), where is an oriented surface with boundary, is associated to a codimension one face of the boundary of the scl norm unit ball in 1 .…”

Scl in graphs of groups

“…(5) Amalgams of free abelian groups, by Susse [32]. (6) t-alternating words in Baumslag-Solitar groups, by Clay-Forester-Louwsma [22].…”

Immersed surfaces in the modular orbifold

“…In [5] it was observed experimentally that for many words w ∈ [F 2 , F 2 ], geodesics on a hyperbolic once-punctured torus corresponding to conjugacy classes of the form [a, b] n w all virtually bound immersed surfaces for sufficiently large n, and it was conjectured (Conjecture 3.16) that this holds in general. Our main theorem (Theorem 3.1 below) proves the natural analogue of this conjecture with the free group F 2 replaced by the virtually free group PSL(2, Z).…”

“…The purpose of this paper is to prove the following Stability Theorem: This theorem proves the natural analogue of Conjecture 3.16 from [5], with PSL(2, Z) in place of the free group F 2 .…”

“…Such questions arise (for example) in topology, complex analysis, contact geometry and string theory. In [5] it was shown that studying isometric immersions between hyperbolic surfaces with geodesic boundary gives insight into the polyhedral structure of the second bounded cohomology of a free group (through its connection to the stable commutator length norm, defined in §2. 2).…”