Gaps in scl for Amalgamated Free Products and RAAGs

TL;DR

This paper establishes a new criterion for detecting the maximal gap in stable commutator length for certain groups, and constructs explicit extremal quasimorphisms to prove these bounds, with applications to amalgamated free products and RAAGs.

Contribution

It introduces a novel method to identify the maximal gap in scl and constructs explicit extremal quasimorphisms for these groups, including RAAGs and groups acting on cube complexes.

Findings

Elements outside conjugates in factors have scl ≥ 1/2.

Non-trivial elements in RAAGs have scl ≥ 1/2, which is sharp.

Constructed explicit extremal quasimorphisms with geometric actions.

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.