Quasi-retracts of groups

TL;DR

This paper explores quasi-retracts of groups, establishing algebraic, geometric properties, and applications, including stability of certain properties under group extensions and relations to hyperbolic structures.

Contribution

It introduces a theory of quasi-split short exact sequences and connects quasi-homomorphisms with quasi-actions, providing new insights into geometric group theory.

Findings

Normal quasi-retracts inherit cobounded actions on hyperbolic spaces

Properties (QFA), (QT'), (PH') are stable under left quasi-split extensions

Quasi-isomorphic groups have isomorphic hyperbolic structures

Abstract

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.