#### Volume 1 (1998)

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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
##### 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.

##### Keywords
groups, adjunction problem, equations over groups, shapes, Magnus derivative, blowing up, amenability
##### Mathematical Subject Classification
Primary: 20E34, 20E22
Secondary: 20E06, 20F05
##### Publication
Received: 15 November 1997
Published: 22 October 1998
##### Authors
 Roger Fenn School of Mathematical Sciences Sussex University Brighton BN1 9QH United Kingdom Colin Rourke Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom