2018/02/04 by Heuer, Nicolaus · 1 citation
#20F36 #20F65 #20F67 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1802.01107
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 ⋆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 non-trivial 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)≤ 1. Such maps were previously unknown, even for non-abelian free groups. For these quasimorphisms ϕ there is an action ρ: G → Homeo+(S1) on the circle such that [δ1 ϕ]=ρ^*euℝb ∈ H2b(G,ℝ), for eu^ℝb the real bounded Euler class.