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

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

1998-10-22

arxiv.org/abs/math/9810184v1

Geom. Topol. Monogr. 1 (1998), 159-166

Mathematics

Group Theory

8 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper8.abs.html

Scientific paper

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.

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