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⋆CB$${G = A \star_C B}$$ we show that every element g in the commutator subgroup of G which does not conjugate into A or B satisfies scl(g)≥1/2$${{\rm scl}(g) \geq 1/2}$$, 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 scl(g)≥1/2$${{\rm scl}(g) \geq 1/2}$$. 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 ϕ¯:G→R$${\bar{\phi} : G \to \mathbb{R}}$$ satisfying ϕ¯(g)≥1$${\bar{\phi}(g) \geq 1}$$ and D(ϕ¯)≤1$${D(\bar{\phi})\leq 1}$$. Such maps were previously unknown, even for non-abelian free groups. For these quasimorphisms ϕ¯$${\bar{\phi}}$$ there is an action ρ:G→Homeo+(S1)$${\rho : G \to \mathrm{Homeo}^+(S^1)}$$ on the circle such that [δ1ϕ¯]=ρ∗eubR∈Hb2(G,R)$${[\delta^1 \bar{\phi}]=\rho^*{\rm eu}^\mathbb{R}_b \in {\rm H}^2_b(G,\mathbb{R})}$$, for eubR$${{\rm eu}^\mathbb{R}_b}$$ the real bounded Euler class.