Kervaire–Laudenbach conjecture

Prove that every nonsingular equation over a group has a solution in some overgroup, thereby resolving the Kervaire–Laudenbach conjecture.

Background

The paper distinguishes nonsingular equations, for which the sum of the exponents of the unknown is nonzero, from unimodular equations, for which that sum is ±1. It notes that certain singular equations have no solution in any overgroup, whereas the existence of a nonsingular equation with the same obstruction is unresolved and is known as the Kervaire–Laudenbach conjecture.

References

The existence of a nonsingular equation with such property is a well-known open question — the Kervaire–Laudenbach conjecture;

— Solvability of unimodular equations in groups and Lie algebras  (2608.28045 - Klyachko et al., 28 Aug 2026) in Section 1, Introduction

it was shown that any unimodular equation over torsion-free group has a solution in some overgroup; while it is unknown whether a similar assertion holds for arbitrary nonsingular equations.

— Solvability of unimodular equations in groups and Lie algebras  (2608.28045 - Klyachko et al., 28 Aug 2026) in Section 1, Introduction