From letter-quasimorphisms to angle structures and spectral gaps for scl

We give a new geometric proof of a theorem of Heuer showing that, in the presence of letter-quasimorphisms (which are analogues of real-valued quasimorphisms with image in free groups), and in particular in RAAGs, there is a sharp lower bound of 1/2 for stable commutator length. Our approach is to show that letter-quasimorphisms give rise to negatively curved angle structures on admissible surfaces. This generalises Duncan and Howie’s proof of the 1/2-lower bound in free groups, and can also be seen as a version of Bavard duality for letter-quasimorphisms.