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Characterisation of a class of equations with solutions
over torsion-free groups
Roger Fenn and Colin Rourke
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Geometry & Topology Monographs 1
(1998) 159–166
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Abstract
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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.
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Keywords
groups, adjunction problem, equations
over groups, shapes, Magnus derivative, blowing up,
amenability
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Mathematical Subject Classification
Primary: 20E22, 20E34
Secondary: 20E06, 20F05
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Publication
Received: 15 November 1997
Published: 22 October 1998
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