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We study equations over torsion-free groups in terms of their
“–shape” (the occurences
of the variable
in the
equation). A
–shape
is
good if any equation with that shape has a solution. It is an outstanding conjecture that
all
–shapes
are good. In a previous article, we proved the conjecture for a large class of
–shapes
called
amenable. Clifford and Goldstein characterised a class of good
–shapes using a
transformation on
–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
–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