Solving equations of length seven over torsion-free groups

Abstract

Prishchepov [16] proved that all equations of length at most six over torsion-free groups are solvable. A different proof was given by Ivanov and Klyachko in [12]. This supports the conjecture stated by Levin [15] that any equation over a torsion-free group is solvable. Here it is shown that all equations of length seven over torsion-free groups are solvable.

On a Nonsingular Equation of Length 9 Over Torsion Free Groups

“…given in [5,8] we conclude that possible vertices of degree 2 (in the diagram associated to P) are (upto cyclic permutation and inversion)…”

“…We now turn our attention to length 9 equations. A list of these equations is given in [3]. Consider the nonsingular equation of length 9 given by atbtctdtetf t −1 gthtit −1 = 1 .…”

“…If G is a torsion free group then by Levin's conjecture every equation is solvable [11]. The conjecture is known to be true for n ≤ 7 [5,7,9,12]. The authors have done significant work in [1,2,4] to establish the conjecture for n = 8.…”

“…The equation of length 9 have been consider in [3]. It has been proved that there are only three equations of length 9 which are still open.…”

“…The following lemma [10] tells us that we can apply asphericity test in k−steps. For a detailed account on the curvature distribution method see [3]. It is clear from our definition of a group equation that if g i is a coefficient between a negative and a positive power of t than g i is not trivial in G. This fact will be used in all subsequent proofs without reference.…”