Locally Indicable Groups of Finite Mal’tsev Rank

Abstract

Recall that G is called a group of finite Mal’tsev rank r if every finitely generated subgroup of G is a group with r generators and r is the least number with this property. Below by the word ‘rank’ we mean the Mal’tsev rank. The notation and terminology adopted here are borrowed from [1, 2]. It is well known that the class of groups of finite rank is closed under subgroups, homomorphic images, and direct products of finitely many groups. For torsion-free Abelian groups, the standard concept of the rank of an Abelian group coincides with that of the Mal’tsev rank. We are also well aware that every torsion-free locally nilpotent group of finite rank is nilpotent and its nilpotency class does not exceed its rank. Groups of finite rank with additional restrictions were studied by many mathematicians such as S. N. Chernikov, V. S. Charin, and D. I. Zaitsev. Research on groups of finite rank was reduced, as a rule, to imposing on a group conditions close to solvability. An impetus for writing this paper is the question whether every linearly orderable finitely generated group will be solvable. The structure of solvable matrix groups over Q is well known. By a theorem of A. I. Mal’tsev, these are finite extensions of triangular matrix groups over Q. We know that every torsion-free solvable group of finite rank can be represented by matrices over the field of rational numbers [1]. Note also that if the rank of a torsion-free solvable matrix group over Q is finite then it is equal to the rational rank (D. I. Zaitsev). In looking into the above question in detail, it became clear that its resolution reduces to treating a similar question for a wider class of groups, namely, the class of locally indicable groups. Recall that a group G is said to be locally indicable if every nontrivial finitely generated subgroup of G admits a homomorphism onto an infinite cyclic group. This class of groups has interesting group-theoretic properties. In [3], it was proved that every one-relator group having no elements of finite order is locally indicable.