We study equations over torsion-free groups in terms of their
"t–shape" (the occurences of the variable t in the equation).
A t–shape is good if any equation with that shape has a
solution. It is an outstanding conjecture that all t–shapes
are good. In a previous article, we proved the conjecture for a large class of
t–shapes called amenable. Clifford and Goldstein
characterised a class of good t–shapes using a transformation on
t–shapes called the Magnus derivative. In this note we
introduce an inverse transformation called blowing up.
Amenability can be defined using blowing up; moreover the connection
with differentiation gives a useful characterisation and implies that
the class of amenable t–shapes is strictly larger than the class
considered by Clifford and Goldstein.
Keywords
groups, adjunction problem, equations
over groups, shapes, Magnus derivative, blowing up,
amenability
