Gaps in scl for Amalgamated Free Products and RAAGs

Abstract

We develop a new criterion to tell if a group G has the maximal gap of 1/2 in stable commutator length (scl). For amalgamated free products {G = A star_C B} we show that every element g in the commutator subgroup of G which does not conjugate into A or B satisfies {{rm scl}(g) geq 1/2}, provided that C embeds as a left relatively convex subgroup in both A and B. We deduce from this that every non-trivial element g in the commutator subgroup of a right-angled Artin group G satisfies {{rm scl}(g) geq 1/2}. This bound is sharp and is inherited by all fundamental groups of special cube complexes. We prove these statements by constructing explicit extremal homogeneous quasimorphisms {bar{phi}: G to mathbb{R}} satisfying {bar{phi}(g) geq 1} and {D(bar{phi})leq 1}. Such maps were previously unknown, even for non-abelian free groups. For these quasimorphisms {bar{phi}} there is an action {rho : G tomathrm{Homeo}^+(S^1)} on the circle such that {[delta^1bar{phi}]=rho^*{rm eu}^mathbb{R}_b in {rm H}^2_b(G,mathbb{R})}, for {{rm eu}^mathbb{R}_b} the real bounded Euler class.