On palindromic width of certain extensions and quotients of free nilpotent groups

In [Bardakov and Gongopadhyay, Palindromic width of free nilpotent groups, J. Algebra 402 (2014) 379–391] the authors provided a bound for the palindromic widths of free abelian-by-nilpotent group ANn of rank n and free nilpotent group N n,r of rank n and step r. In the present paper, we study palindromic widths of groups [Formula: see text] and [Formula: see text]. We denote by [Formula: see text] the quotient of the group Gn = 〈x1, …, xn〉, which is free in some variety by the normal subgroup generated by [Formula: see text]. We prove that the palindromic width of the quotient [Formula: see text] is finite and bounded by 3n. We also prove that the palindromic width of the quotient [Formula: see text] is precisely 2(n - 1). As a corollary to this result, we improve the lower bound of the palindromic width of N n,r. We also improve the bound of the palindromic width of a free metabelian group. We prove that the palindromic width of a free metabelian group of rank n is at most 4n - 1.

GEOMETRICAL APPROACH TO THE FREE SOLVABLE GROUPS

We give a topological interpretation of the free metabelian group, following the plan described in [11, 12]. Namely, we represent the free metabelian group with d-generators as an extension of the group of the first homology of the d-dimensional lattice (as Cayley graph of the group ℤd), with a canonical 2-cocycle. This construction opens a possibility to study metabelian groups from new points of view; in particular to give useful normal forms of the elements of the group, leading to applications to the random walks, and so on. We also describe the satellite groups which correspond to all 2-cocycles of cohomology group associated with the free solvable groups. The homology of the Cayley graph can be used for studying the wide class of groups which include the class of all solvable groups.

THE BINARY ADDING MACHINE AND SOLVABLE GROUPS

We prove that any solvable subgroup K of automorphisms of the binary tree, which contains the binary adding machine is an extension of a torsion-free metabelian group by a finite 2-group. If the group K is moreover nilpotent then it is torsion-free abelian.