Final Report Summary - GETEMO (Geometry, Groups and Model Theory.)
In this ERC project, we have studied several mathematical problems at the intersection of group theory, model theory, combinatorics, random walks and number theory and have made significant progress on them. One theme has been the use of the model-theoretic point of view to tackle certain combinatorial problems, for example in Erdos geometry, which in turn have had applications to the study of finite simple groups and their Cayley graphs. In this vein, we have furthered the study of approximate groups initiated more than a decade ago in order to study the spectrum of finite Cayley graphs with number theoretical applications in mind (affine sieve, thin groups). We have studied the relationship between spectral properties of infinite groups and their finite quotients, settling a long standing open problem in the infinite dimensional representation theory of discrete groups. We have also studied these problems in the simpler context of solvable groups, making explicit a connection with diophantine number theory and the Lehmer problem in particular. In doing so, we have settled long standing conjectures pertaining to random polynomials of large degree and Bernoulli convolutions. In relation to these non-commutative diophantine problems we have also made new progress on metric diophantine approximation on manifolds as well as on the study of word maps in Lie groups and their associated character varieties. Finally we have supervised several doctoral theses on related topics.