#### 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.

Notes on file formats
Roger Fenn, Colin Rourke

School of Mathematical Sciences, Sussex University

Brighton, BN1 9QH, UK

Mathematics Institute, University of Warwick

Coventry, CV4 7AL, UK

Email: R.A.Fenn@sussex.ac.uk, cpr@maths.warwick.ac.uk

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