Equations with torsion-free coefficients

Abstract

In this paper we generalize techniques used by Klyachko and the authors to prove some tessellation results about S 2 . These results are applied to prove the solvability of certain equations with torsion-free coefficients.

Free subgroups of one-relator relative presentations

“…If the exponent sum of the generator in the word w equals to any number p = ±1, then it is unknown even whether the group G embeds into G = G, t | w = 1 ; in other words, it is unknown when the group G is different from Z/pZ. There are a lot of papers on this subject, but the answer is known only under additional strong restrictions on the group G or/and on the word w (see, e.g., [B84], [KP95], [C02], [C03], [CG95], [CG00], [EH91], [FeR98], [GR62], [IK00], [Le62], [Ly80], [S87]). For this reason, in this paper we study only unimodular presentations.…”

The Kervaire-Laudenbach Conjecture and Presentations of Simple Groups

“…The following lemma is a version of an algebraical trick in [4], which is often used in studying equations over groups, and related issues (see [5][6][7][8][9][10][11]). Its geometrical interpretation can be found in [6].…”

A Class of Exponent-Sum Two Equations Over Groups

“…Following [3] and [4], we will add a family of dotted edges to À. Let Á be any region of À whose boundary is not consistently directed.…”

“…It is a consequence of work done in [4], that if k is not solvable over the torsion free group G, then there is some positive word in either fa 1 ; b À1 1 g or fa k ; b À1 k g that represents the identity in G. We see that Theorem 2 implies that for each i, there is a positive word in fa i ; b À1 i g that represents the identity in G. Therefore, Theorem 2 is a stronger result for k than that given in [4].…”