On the residual finiteness of the non-abelian exterior square of wreath products and the Grigorchuk group

We show some sufficient conditions ( in terms of properties of $A$, $B$ and $X$) for the non-abelian exterior square $G \wedge G$ to be residually finite, where $G $ is the wreath product $A \wr B$ or the combinatorial wreath product $G = A \wr_X B$, in the latter case $B$ is a free group. We prove