A Hurewicz-type formula for asymptotic-dimension-lowering symmetric quasimorphisms of countable approximate groups

A well-known Hurewicz-type formula for asymptotic-dimension-lowering group homomorphisms, due to A. Dranishnikov and J. Smith, states that if f:G→ H is a group homomorphism, then asdim G ≤ asdim H + asdim ( f). In this paper we establish a similar formula for certain quasimorphisms of countable approximate groups: if (Ξ, Ξ^∞) and (Λ, Λ^∞) are countable approximate groups and if f:(Ξ, Ξ^∞)→ (Λ,Λ^∞) is a symmetric unital quasimorphism, we show that asdim Ξ ≤ asdim Λ + asdim (f⁻¹\!(D(f))), where $D(f)$ is the defect set of f.