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Characterisation of a class of equations with solutions over torsion-free groups

Roger Fenn and Colin Rourke

Geometry & Topology Monographs 1 (1998) 159–166

DOI: 10.2140/gtm.1998.1.159

arXiv: math.GR/9810184


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.


groups, adjunction problem, equations over groups, shapes, Magnus derivative, blowing up, amenability

Mathematical Subject Classification

Primary: 20E22, 20E34

Secondary: 20E06, 20F05


Received: 15 November 1997
Published: 22 October 1998

Roger Fenn
School of Mathematical Sciences
Sussex University
United Kingdom
Colin Rourke
Mathematics Institute
University of Warwick
United Kingdom