The Kervaire conjecture and the minimal complexity of surfaces

Abstract:

Talk 1

Title: Weights of groups

Abstract: This is an introductory talk on weights of groups. The weight (also called the normal rank) of a group \(G\) is the smallest number of elements that normally generate \(G\). We will discuss basic properties and examples in connection to topology. Although it is a simple notion, several basic problems remain open, including the Kervaire conjecture and the Wiegold question. We will explain some well-known partial results and their proofs.

Talk 2

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, which has connection to various problems in topology. 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, i.e. the group \(G*Z\) has weight greater than 1. We interpret this as a problem of estimating the minimal complexity (in terms of Euler characteristic) of surface maps to certain spaces. This gives a conceptually simple proof of Klyachko's theorem that confirms the Kervaire conjecture for any \(G\) torsion-free. We also obtain injectivity of the map \(G\to(G*Z)/<<w>>\) when \(w\) is a proper power for arbitrary \(G\). Both results generalize to certain HNN extensions.

algebraic topologyfunctional analysisgroup theorygeometric topologyoperator algebras

Audience: researchers in the topic

Vienna Geometry and Analysis on Groups Seminar