Stable Commutator Length in Baumslag–solitar Groups and Quasimorphisms for Tree Actions

This paper has two parts, on Baumslag–Solitar groups and on general G–trees. In the first part we establish bounds for stable commutator length (scl) in Baumslag–Solitar groups. For a certain class of elements, we further show that scl is computable and takes rational values. We also determine exactly which of these elements admit extremal surfaces. In the second part we establish a universal lower bound of 1/12 for scl of suitable elements of any group acting on a tree. This is achieved by constructing efficient quasimorphisms. Calculations in the group BS(2, 3) show that this is the best possible universal bound, thus answering a question of Calegari and Fujiwara. We also establish scl bounds for acylindrical tree actions. Returning to Baumslag–Solitar groups, we show that their scl spectra have a uniform gap: no element has scl in the interval (0, 1/12).