Title: The Kervaire conjecture and the minimal complexity of surfaces

Abstract: We use topological methods to solve special cases of a fundamental problem in group theory, the Kervaire conjecture.
The conjecture asserts that, for any nontrivial group G and any element w in the free product G*Z, the quotient (G*Z)/<<w>> is still nontrivial. We interpret this as a problem of estimating the minimal complexity (in terms of Euler characteristic) of surfaces in HNN extensions. This gives a conceptually simple proof of Klyachko’s theorem that confirms the Kervaire conjecture for any G torsion-free. I will also explain new results obtained using this approach.