Geometry & Topology Monographs 1 (1998), The Epstein Birthday Schrift, paper no. 8, pages 159-166.

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

Roger Fenn, Colin Rourke

Abstract. 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 [Klyachko's methods and the solution of equations over torsion-free groups, l'Enseign. Maths. 42 (1996) 49--74] we proved the conjecture for a large class of t-shapes called amenable. In [Tesselations of S^2 and equations over torsion-free groups, Proc. Edinburgh Maths. Soc. 38 (1995) 485--493] 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

AMS subject classification. Primary: 20E34, 20E22. Secondary: 20E06, 20F05.

E-print: arXiv:math.GR/9810184

Submitted: 15 November 1997. Published: 22 October 1998.

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Roger Fenn, Colin Rourke
School of Mathematical Sciences, Sussex University
Brighton, BN1 9QH, UK
Mathematics Institute, University of Warwick
Coventry, CV4 7AL, UK

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