The aim of these notes is to provide an introduction to Gaboriau’s paper « Sur le coût des relations d’équivalence et des groupes » and to gather in one place various arguments, occuring in diverse sources, to give a characterisation of amenable groups through their actions on probability spaces.
Category Archives: Cost
Notes on the Abért–Nikolov theorem on rank gradient and cost (notes by Holger Kammeyer after his own lecture)
These are notes of a talk on a theorem of Abért–Nikolov: Let \(\Gamma\) be a finitely generated group and let \((\Gamma_n)\) be a chain of finite index subgroups. Assume that the action of \(\Gamma\) on the boundary \(\partial T\) of the coset tree of \((\Gamma_n)\) is essentially free. Then the rank gradient of \(\Gamma\) with respect to the chain \((\Gamma_n)\) equals the cost of the action of \(\Gamma\) on \(\partial T\).