2025/01/04 by Wan, Renxing
#20F65 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2501.02238
In this paper, we study a special class of quasi-homomorphisms, i.e. quasi-retractions from a group to its subgroups. We first give some algebraic and geometric properties of quasi-retracts and then propose a theory of quasi-split short exact sequences of groups. Later, we establish a connection between quasi-homomorphisms and induced quasi-actions. Finally, we give some geometric applications of quasi-homomorphisms, including normal quasi-retracts inherit cobounded actions on hyperbolic spaces, properties (QFA), (QT') and (PH') are all stable under left quasi-split group extensions, quasi-isomorphic groups have isomorphic hyperbolic structures, and so on.