Gaps in scl for Amalgamated Free Products and RAAGs

Nicolaus Heuer

doi:10.1007/s00039-019-00477-5

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.

Highlights

For a group G let G be the commutator subgroup
For amalgamated free products G = A 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, provided that C embeds as a left relatively convex subgroup in both A and B
We deduce from this that every nontrivial element g in the commutator subgroup of a right-angled Artin group G satisfies scl(g) ≥ 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 satisfying φ(g) ≥ 1 and D(φ) ≤

Summary

Introduction

For a group G let G be the commutator subgroup. For an element g ∈ G the commutator length (cl(g)) denotes the minimal number of commutators needed to express g as their product. By Bavard’s Duality Theorem we infer that any such g which lies in the commutator subgroup satisfies scl(g) ≥ 1/2 We apply this to right-angled Artin groups using the work of [ADS15]. Every non-trivial element g ∈ G in the commutator subgroup of a right-angled Artin group G satisfies scl(g) ≥ 1/2. We obtain gap results even for elements which are not in the commutator subgroup This suggests that it may be interesting to use Bavard’s Dualtiy Theorem as a generalisation of stable commutator length to an invariant of general group elements g ∈ G. We show the 1/2 gaps for right-angled Artin groups in Section 7; see Theorem 7.3

Quasimorphisms and Bavard’s Duality Theorem

Letter-Thin Triples and the Maps α and β

Gaps via Letter-Quasimorphisms

Left Orders and Left-Relatively Convex Subgroups

Amalgamted Free Products

Right-Angled Artin Groups