The solution of length three equations over groups

Abstract

Let G be a group, and let r = r{t) be an element of the free product G * <£> of G with the infinite cyclic group generated by t. We say that the equation r(i) = 1 has a solution in G if the identity map on G extends to a homomorphism from G * <£> to G with r in its kernel. We say that r(t) = 1 has a solution over G if G can be embedded in a group H such that r(t) = 1 has a solution in H. This property is equivalent to the canonical map from G to <G, t | r> (the quotient of G * <t> by the normal closure of r) being injective.In general it is not possible to find a solution to an arbitrary equation r(t) = 1 over an arbitrary group G. It is necessary to place some sort of restriction on the group, on the equation, or possibly on both. One possible restriction on the equation is that the exponent sum of t in r be non-zero. Under this hypothesis, but with no restriction on the group G, it is an open problem whether a solution over G always exists.It is known that a solution exists if G is either locally residually finite [6] or locally indicable [1,3,9], and other known results give solutions under restrictions on r. Levin [4] showed that a solution exists if t occurs in r only with positive exponent. Thus the simplest remaining case (up to conjugacy and inversion) is when r(t) has the form atbtct' 1 (a,b,ceG). Lyndon [5, Corollary 5.3] has solved this case under certain restrictions on the "coefficients" a, b, c. These restrictions are based on small cancellation theory, and concern the relations which can hold in G between a, b and c.It is the purpose of this note to remove the restrictions from Lyndon's result, and show that any equation of the form over any group G has a solution over G. Combined with Levin's theorem, this solves the problem whenever t occurs at most 3 times in r(t).Note that the above equation can be transformed to one of the form by applying the automorphism g>->g feeG), tr-ttb' 1 of G*(ty, so we are reduced to the case b=\. Also, if the equation has a solution over the subgroup G o of G generated by *Research supported by a William Gordon Seggie Brown Fellowship. 89

Tesselations ofS2and equations over torsion-free groups

“…The following lemma is just the dual situation to a well-known result of Howie [4], and shall be stated without proof. Proof.…”

Non-Amenable Type K Equations Over Groups

“…Since Howie used relative diagrams in [11] to prove that all length three equations are solvable over all groups, it has become standard to apply results about tesselations of the 2-sphere to the study of equations over groups. This was Klyachko's approach: to translate his geometric result into a group-theoretic theorem.…”

“…That is to say, if G is a non-trivial torsion-free group and e ∈ G * t , then G, t | e is not trivial. (It is not hard to see that this question is only interesting in the case where t has exponent sum 1 in e.)Since Howie used relative diagrams in [11] to prove that all length three equations are solvable over all groups, it has become standard to apply results about tesselations of the 2-sphere to the study of equations over groups. This was Klyachko's approach: to translate his geometric result into a group-theoretic theorem.…”

“…General results. In [11], Howie used relative diagrams to show that equations satisfying |e| = 3 are solvable over all groups. Since then, many authors have used relative diagrams, and the dual notion of relative pictures, to investigate equations over groups.…”

Equations with torsion-free coefficients

“…Since Howie [7] introduced them, relative diagrams have been used by many authors to discuss equations over groups. It seems that most reasonably general results about tessellations of the two-sphere can be translated into results about group theory.…”