Equations Over Groups and a Theorem of Higman, Neumann, and Neumann

Abstract

THEOREM. Let G be a group and a,deG such that {\a\, \d\} =£{2, 3}. The equation at n dt~m = 1 (n, m e Z, n # m) has a solution over G. A.M.S. (1980) subject classification: 20E06.

Tesselations ofS2and equations over torsion-free groups

“…Combinatorial results such as this were anticipated by Stallings in [8, p. 147]. There seem to be two techniques for proving facts concerning graphs embedded on S 2 ; curvature (weight tests) (see for example [1,2,3,7,8]), and minimal circle techniques (see [5,6]). As it is a generalization of the proof of Klyachko, our proof uses the latter.…”

“…A pattern P is a directed tree embedded in R 2 with a specified vertex, called the centre, which is adjacent to each edge. If we reverse the orientation of each edge of the pattern P and reflect this directed tree in R 2 , we obtain a new pattern denoted P. If we label some of the corners at the centre of P with distinct positive labels taken from some alphabet X, we get a labelled pattern denoted P x . The inverse labelled pattern, P x is obtained from P by labelling the corner corresponding to the corner of P x labelled <x with the label a.…”

Asphericity of a length four relative group presentation

“…To prove asphericity we adopt the approach of [6]. Let each corner in every region of P be given an angle.…”

Equations with torsion-free coefficients

“…It seems that most reasonably general results about tessellations of the two-sphere can be translated into results about group theory. Work following this scheme can be found in [3,4,6,8].…”