The Kervaire conjecture: adjoining one generator and one relator never kills a nontrivial group

Claims Theorem 1.1: for every group $A$ (no torsion, cardinality or presentation hypothesis) and every $w\in A*\langle t\rangle$ with exponent sum $\pm1$ in $t$, the canonical map $A\to(A*\langle t\rangle)/\langle\langle w\rangle\rangle$ is injective, so the equation $w(t)=1$ is solvable in an overgroup. Corollary 1.2: the quotient is nontrivial for every $w$ when $A\ne1$, which is the Kervaire conjecture. It does NOT prove injectivity for other nonzero exponent sums (the Kervaire-Laudenbach form); that is the companion's nonsingular-systems theorem. It does not assert a solution inside $A$ itself.

Claims Theorem 1.1: for every group $A$ (no torsion, cardinality or presentation hypothesis) and every $w\in A*\langle t\rangle$ with exponent sum $\pm1$ in $t$, the canonical map $A\to(A*\langle t\rangle)/\langle\langle w\rangle\rangle$ is injective, so the equation $w(t)=1$ is solvable in an overgroup. Corollary 1.2: the quotient is nontrivial for every $w$ when $A\ne1$, which is the Kervaire conjecture. It does NOT prove injectivity for other nonzero exponent sums (the Kervaire-Laudenbach form); that is the companion's nonsingular-systems theorem. It does not assert a solution inside $A$ itself.