| Résume | For infinite groups, the Følner criterion states that a group is amenable if and only if the isoperimetric constant of its Cayley graph is 0. Its isoperimetric profile can then be described by the Følner function. It depends on the choice of generating set, but different functions on the same group are asymptotically equivalent. Multiple results have been obtained up to asymptotic equivalence class, however more precise descriptions have rarely been studied. In this talk, we will consider fixed generating sets and present (to our knowledge) the first results (outside of virtually nilpotent groups) on the exact values of Følner functions - on wreath products ℤ ≀ D for a finite group D. In a work in progress joint with Nikolay Ivanov, we aim to also obtain a result on Baumslag-Solitar groups BS(1,n). We will also discuss connections with the Coulhon and Saloff-Coste inequality which provides a lower bound on the Følner function. In joint work with Christophe Pittet, for groups of exponential growth we obtain a description of the optimal multiplicative constant in the Coulhon and Saloff-Coste inequality. We show that the optimal value over all groups of this constant is between 1 and 2. |
