Unimodular equations which do not preserve the derived length of a group
Mikhail A. Mikheenko
TL;DR
The paper addresses whether unimodular equations over solvable groups can always be solved within solvable groups of the same derived length. It shows, for every $n\ge2$, a unimodular equation over a solvable group of derived length $n$ with no solution in any solvable group of the same length, by constructing a wreath-product setup $H = B \wr G$ with a carefully chosen metabelian seed $G$ and examining the equation $w(x)=1$ given by $x x^{-a} x^{a^2} = c$. The argument analyzes the solution group's derived series, demonstrating that certain commutator configurations force nontrivial higher derived subgroups, leading to a contradiction and establishing the nonexistence of length-$n$ solvable solutions. This advances our understanding of the limits of unimodular equation solvability in solvable groups and highlights open questions about extending solvability to longer derived lengths and specific groups.
Abstract
It is a known fact that any unimodular equation over an abelian group has a solution in that group itself. It is also known that for metabelian groups this does not hold; moreover, there is a unimodular equation over some metabelian group which has no solutions in any larger metabelian group. Here we present the proof of an analagous fact for solvable groups of higher derived lengths.