Equations over Torsion-free Groups Summary Klyachko's Methods and the Solution of Equations over Torsion-free Groups

SUMMARY The question we are concerned with here is the following : Let G be a torsion-free group and consider the free product Ghti of G with an innnite cyclic group (generator t). Let w be an element of Ghti?G and hhwii denote the normal closure of w in G hti, then is the natural homomorphism G ! G hti hhwii injective? Klyachko's paper : \Funny property of sphere and equations over groups" Kl] contains a proof that it is injective in the case in which the exponent sum of t in w is 1. If the exponent sum is not 1 then Ghti hhwii has a non-trivial cyclic quotient. So the following is implied: Corollary (Kervaire conjecture for torsion-free groups) Let G be a non-trivial torsion-free group then Ghti hhwii is non-trivial. The proof for exponent sum 1 is based on Klyachko's \funny property of sphere". This is the following: Let K be a cell subdivision of the 2{sphere with a least one 1{cell. Let a car drive round the boundary of each 2{cell in an anticlockwise sense (the cars travel at arbitrary speeds, never stop and visit each point of the boundary of the cell innnitely often). Then there must be at least two places on the sphere where complete crashes occur (a complete crash is either a head-on collision in the middle of a 1{cell or a crash at a vertex involving all the cars from neighbouring 2{cells). Klyachko describes this property as \suitable for a school mathematics tourna-ment". The property is used to show that the diagram for a potential counterexample to the Kervaire conjecture must have at least one interior vertex with all labels being the same element of G, hence this element has nite order. In this paper we shall give an exposition of Klyachko's methods and theorems. We use his techniques to give a positive answer to the question for other exponents under a technical condition on the t{shape of w , for details here see section 5.