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GETEMO Report Summary

Project ID: 617129
Funded under: FP7-IDEAS-ERC
Country: Germany

Periodic Report Summary 3 - GETEMO (Geometry, Groups and Model Theory.)

The research related to this project has been devoted to number of important open problems in pure mathematics and in particular in group theory and number theory. We have approached these problems using novel methods and recently developed tools coming from diophantine analysis, combinatorics and mathematical logic.

One important aspect of our work focused on the study of Cayley graphs of finite groups, describing the intimate relationship that exists between their geometry, the nature of the spectrum and the algebraic properties of the groups considered. We have exploited the recently uncovered structure theory of approximate groups in order to give uniform lower bounds on the spectrum of finite simple groups and upper bound on their diameter.

Another main direction undertaken in this ERC project has been revolving around the Lehmer conjecture from diophantine geometry. We gave two different reinterpretations of this conjecture, one in terms of a general lower bound on the spectrum of a certain class of hyperbolic surfaces of arithmetic origin, and another in terms of a lower bound on the growth of subgroups of linear transformations of the complex plane.
In parallel and jointly with P. Varjù we have began to further exploit combinatorial tools in relation to discretized versions of the sum-product phenomenon and its ramifications for approximate subgroups of Lie groups, in order to study Bernoulli convolutions and fine properties of random walks on Lie groups. In particular we have been led to solve a important open problem in metric diophantine approximation and homogeneous dynamics about submanifolds of matrices and used it to establish exact computations of diophantine exponents for certain non-commutative Lie groups.

In parallel I have supervised four graduate students who did remarkable work on several important related problems using the tools and methods discussed above. For example a large deviation principle for random matrix products was established by Cagri Sert, a high-dimensional discretized sum-product and projection theorem was worked out by Weikun He and a limit multiplicity theorem for congruence arithmetic surfaces and 3-manifolds was proven by Mikolaj Fraczyk. I also supervised and had a fruitful collaboration with two post-docs Matthew Tointon and Mohammad Bardestani. I now have many on going subprojects that will undoubtedly lead to further challenging prospects.

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