POLYCYCLIC, METABELIAN OR SOLUBLE OF TYPE (FP)∞ GROUPS WITH BOOLEAN ALGEBRA OF RATIONAL SETS AND BIAUTOMATIC SOLUBLE GROUPS ARE VIRTUALLY ABELIAN

VITALY ROMAN'KOV
2017 Glasgow Mathematical Journal  
Let G be a polycyclic, metabelian or soluble of type (FP)∞ 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.
doi:10.1017/s0017089516000677 fatcat:jjde5ueqinhnpmk55gxnrqx4ny