The Solution of Length Five Equations Over Groups

Abstract

Let G be a group, t an unknown, and r(t) an element of the free product G* ⟨ t⟩. The equation r(t) = 1 has a solution over G if it has a solution in a group H containing G. The Kervaire–Laudenbach (KL) conjecture asserts that if the exponent sum of t in r(t) is nonzero the equation has a solution. Equations of length 5 have been studied and it was proved that a solution exists under certain restrictions imposed on the coefficients of the equation. This article removes these restrictions and therefore settles the KL conjecture for equations of length five.

Key Words:

• Equation
• Group
• Relative diagrams

Mathematics Subject Classification:

• Primary 20E06
• Secondary 20F06

Notes

Communicated by A. Yu. Olshanskii.