On the residual finiteness of the non-abelian exterior square of some Artin groups, constructible soluble groups, generalised Baumslag-Solitar groups and $SL_n(\Z)$

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 $FP_{\infty}$ we prove that the groups $\X(G)$, $ν(G)$, $G \wedge G$ are virtually residually $p$-finite for almost all primes $p$. For $G$ a generalised Baumslag-Solitar group of rank $n \geq 1$ that is residually finite, we show that $G \wedge G$ is residually finite. For $G = SL_n(\Z)$ and $G = GL_n( \mathbb{Z})$, $n \geq 3$, we show that $G \wedge G$ is residually finite.