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 $scl(g) \geq 1/2$, 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 $scl(g) \geq 1/2$. 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 $\rho : G \to Homeo^+(S^1)$ on the circle such that $[\delta^1 \bar{ \phi}]=\rho^*eu^{\mathbb{R}}_b \in H^2_b(G,\mathbb{R})$, for $eu^\mathbb{R}_b$ the real bounded Euler class.