Obiettivo The goal of the project is to develop new techniques for estimation and evaluation of well-known asymptotic invariants of groups, including growth of groups, isoperimetric profiles, entropy and probability to return to the origin of random walks as well as of some more recent invariants related to the geometric criteria for construction of quotients of groups, which appeared in the joint work of PI with A.Karlsson (2010), and in a recent work of Ozawa (2015) giving a short functional analytic proof of the Polynomial Growth Theorem. We plan to work on the Gap conjecture of Grigorchuk, which states that any group of growth asymptotically strictly less than exp(n1/2) has polynomial growth, the question about the forms of Foelner sets in groups of intermediate and exponential growth and Kaimanovich and Vershik conjecture about characterisation of groups of exponential growth in terms of non-triviality of the Poisson boundary of some symmetric random walks. We plan to develop methods sharpening previous results of PI about isoperimetric inequalities for wreath products and relation between growth of groups and isoperimetry, and apply them for growth estimates and the description of the boundary. A further goal of the project is to introduce new constructions of non-elementary amenable groups which can show that necessary conditions in growth conjecture and isoperimetric inequalities cannot be weakened. Parole chiave Growth of Groups Groups of Intermediate Growth Isoperimetric inequalities entropy random walks on groups Poisson boundary Grigorchuk conjecture Grigorchuk groups Programma(i) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Argomento(i) ERC-2016-COG - ERC Consolidator Grant Invito a presentare proposte ERC-2016-COG Vedi altri progetti per questo bando Meccanismo di finanziamento ERC-COG - Consolidator Grant Istituzione ospitante CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS Contribution nette de l'UE € 1 789 438,00 Indirizzo RUE MICHEL ANGE 3 75794 Paris Francia Mostra sulla mappa Regione Ile-de-France Ile-de-France Paris Tipo di attività Research Organisations Collegamenti Contatta l’organizzazione Opens in new window Sito web Opens in new window Partecipazione a programmi di R&I dell'UE Opens in new window Rete di collaborazione HORIZON Opens in new window Costo totale € 1 789 438,00 Beneficiari (1) Classifica in ordine alfabetico Classifica per Contributo netto dell'UE Espandi tutto Riduci tutto CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS Francia Contribution nette de l'UE € 1 789 438,00 Indirizzo RUE MICHEL ANGE 3 75794 Paris Mostra sulla mappa Regione Ile-de-France Ile-de-France Paris Tipo di attività Research Organisations Collegamenti Contatta l’organizzazione Opens in new window Sito web Opens in new window Partecipazione a programmi di R&I dell'UE Opens in new window Rete di collaborazione HORIZON Opens in new window Costo totale € 1 789 438,00