Quasi-affine and quasi-quadratic maps of groups with non-abelian targets

TL;DR

This paper explores the structure and rigidity of quasi-homomorphisms and quasi-quadratic maps in group theory, revealing their precise perturbation properties and establishing their constructibility and rigidity in hyperbolic groups.

Contribution

It characterizes middle quasi-homomorphisms as constant perturbations of quasi-homomorphisms and introduces quasi-polynomial maps, demonstrating their rigidity in hyperbolic groups.

Findings

Middle quasi-homomorphisms are constant perturbations of quasi-homomorphisms

Quasi-polynomial maps are constructible

Quasi-quadratic maps into hyperbolic groups are rigid under bounded perturbations

Abstract

It is shown that the middle quasi-homomorphisms of Fujiwara and Kapovich are precisely constant perturbations of quasi-homomorphisms. Quasi-polynomial maps are defined and their constructibility is explored. In particular, it is shown that a large class of quasi-quadratic maps into torsion-free hyperbolic groups is rigid with respect to bounded perturbations.