The Kervaire conjecture and the minimal complexity of surfaces

Abstract

The Kervaire conjecture asserts that adding a generator and then a relator to a nontrivial group always results in a nontrivial group. We introduce new methods from stable commutator length to study this type of problems about nontriviality of one-relator quotients. Roughly, we show that surfaces in certain HNN extensions bounding a given word have complexity no less than the complexity of its boundary. A consequence of this is a Freiheitssatz theorem for HNN extensions, which in particular implies and gives a new proof of Klyachko's theorem that confirms the Kervaire conjecture for torsion-free groups. As another application, we also generalize the following theorem of Klyachko-Lurye to HNN extensions: For any group $G$ and the quotient $Q$ of $G\star\mathbb{Z}$ by any proper power $w^m$ with $w\in G\star\mathbb{Z}$ projecting to $1\in\mathbb{Z}$, the natural map $G\to Q$ is injective.