Equations Over Groups

Let G be a group, and let r = r(t) be an element of the free product G * 〈G〉 of G with the infinite cyclic group generated by t. We say that the equation r(t) = 1 has a solution in G if the identity map on G extends to a homomorphism from G * 〈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 * 〈G〉 by the normal closure of r) being injective.

Download Full-text

The Solution of Length Five Equations Over Groups

Communications in Algebra ◽

10.1080/00927870701247039 ◽

2007 ◽

Vol 35 (6) ◽

pp. 1914-1948 ◽

Cited By ~ 2

Author(s):

Anastasia Evangelidou

Keyword(s):

Equations Over Groups

Download Full-text

Reducible Diagrams and Equations Over Groups

Essays in Group Theory - Mathematical Sciences Research Institute Publications ◽

10.1007/978-1-4613-9586-7_2 ◽

1987 ◽

pp. 15-73 ◽

Cited By ~ 27

Author(s):

S. M. Gersten

Keyword(s):

Equations Over Groups

Download Full-text

A new test for asphericity and diagrammatic reducibility of group presentations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.126 ◽

2019 ◽

Vol 150 (2) ◽

pp. 871-895 ◽

Cited By ~ 1

Author(s):

Jonathan Ariel Barmak ◽

Elias Gabriel Minian

Keyword(s):

Weak Version ◽

Zero Divisors ◽

Group Presentations ◽

Equations Over Groups ◽

Weight Test

AbstractWe present a new test for studying asphericity and diagrammatic reducibility of group presentations. Our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails. We use this criterion to generalize results of J. Howie and S.M. Gersten on asphericity of LOTs and of Adian presentations, and derive new results on solvability of equations over groups. We also use our methods to investigate a conjecture of S.V. Ivanov related to Kaplansky's problem on zero divisors: we strengthen Ivanov's result for locally indicable groups and prove a weak version of the conjecture.

Download Full-text