On the residual finiteness of the non-abelian exterior square of some Artin groups, constructible soluble groups, generalised Baumslag-Solitar groups and $S{L}_n(\mathbb{Z})$

Abstract

We show that cohomological and homological goodness are equivalent. We prove that if $G$ is a coherent even Artin group or an even Artin group whose underlying graph does not contain 4-clique or a LERF Artin group then $G\otimes G$ and $G\wedge G$ are residually finite. Furthermore for $G$ a RAAG both $G\otimes G$ and $G\wedge G$ are residually $p$-finite for any prime integer $p$. For $G$ a limit group or a Coxeter group we show that $G\wedge G$ is residually finite. For $G$ a soluble group of type $F{P}_{\infty }$ we prove that the groups $\X(G)$, $\nu (G)$, $G\wedge G$ are virtually residually $p$-finite for almost all primes $p$. For $G$ a generalised Baumslag-Solitar group of rank $n\ge 1$ that is residually finite, we show that $G\wedge G$ is residually finite. For $G=S{L}_n(\mathbb{Z})$ and $G=G{L}_n(\mathbb{Z})$, $n\ge 3$, we show that $G\wedge G$ is residually finite.