$\gamma$-Chiral is same as Chiral

Abstract

A word $w$ in a free group is called {\em chiral} if there exists a group $G$ such that image of word map corresponding to word $w$ is not closed with respect to inverse. Similarly a group $G$ is said to be {\em chiral} if there exists a word $w$ in free group such that $w$ exhibits chirality on the group $G$. Gordeev et al. \cite{gordeev2018geometry} extended the concept of chirality to introduce $\gamma$-chirality in both cases. We show that the notion of $\gamma$-chirality is equivalent to chirality.