Middle quasi-homomorphisms are precisely constant perturbations of quasi-homomorphisms, and nearly normal quasi-quadratic maps into torsion-free hyperbolic groups are rigid under bounded perturbations.
Gowers, A new proof of Szemer\' edi's theorem for arithmetic progressions of length four , Geom
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Quasi-affine and quasi-quadratic maps of groups with non-abelian targets
Middle quasi-homomorphisms are precisely constant perturbations of quasi-homomorphisms, and nearly normal quasi-quadratic maps into torsion-free hyperbolic groups are rigid under bounded perturbations.