Polycyclic, metabelian or soluble of type (FP)$_{\infty}$ groups with boolean algebra of rational sets and biautomatic soluble groups are virtually abelian

Abstract

Let $G$ be a polycyclic, metabelian or soluble of type (FP)$_{\infty}$ group such that the class $Rat(G)$ of all rational subsets of $G$ is a boolean algebra. Then $G$ is virtually abelian. Every soluble biautomatic group is virtually abelian.