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

TL;DR

This paper proves that certain classes of groups, including polycyclic, metabelian, and soluble groups with specific properties, are virtually abelian, and also shows that all soluble biautomatic groups are virtually abelian.

Contribution

It establishes that groups with a boolean algebra of rational subsets are virtually abelian and proves that all soluble biautomatic groups are virtually abelian.

Findings

Groups with boolean algebra of rational subsets are virtually abelian

All soluble biautomatic groups are virtually abelian

The results unify properties of rational sets and automatic groups in group theory

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.