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 that for the Grigorchuk group $G$ the non-abelian exterior square $G\wedge G$ and the non-abelian tensor product $G\otimes G$ are residually finite. We show conditions that imply that $A\wr B$ is not (cohomologically) good in dimension $\le 2$.