Quasi-retracts of groups

Abstract: A subgroup $H\le G$ is a quasi-retract of $G$ if there exists a quasi-homomorphism from $G$ to $H$ which restricts to the identity on $H$. In this paper, we study under what conditions a subgroup is a quasi-retract of the whole group. We also show that a subgroup of $F_2$ is a nontrivial quasi-retract if and only if it is cyclic. As an intermediate result, we show that for any short exact sequence of groups $1\to H\to G\to G/H\to 1$, the following are equivalent: (i) it is left quasi-split (or equivalently, $H$ is a quasi-retract of $G$). (ii) it is right quasi-split and $H$ almost commutes with a right coset transversal of $H$ in $G$. (iii) $G$ is strictly quasi-isomorphic to $H\times G/H$. This is an analogue to the classical result on split short exact sequence of groups.