### Balls in groups: volume, structure and growth

Geometry and Topology Seminar

7th May 2024, 2:00 pm – 3:00 pm

Fry Building, 2.04

One of Gromov's most famous theorems states that a group with polynomial growth is virtually nilpotent. Recently "finitary" refinements of this theorem have been proved by Shalom-Tao and Breuillard, Green and Tao. Roughly those statements are of the following form: assuming a polynomial upper bound V(r) < Cr^d on the volume of a ball of fixed, but large radius r, they deduce that the group is "not too far" from being nilpotent. With Matthew Tointon, we give sharp bounds in Breuillard, Green and Tao's version of Gromov's theorem. Precisely, the bounds on both the nilpotence and index are sharp; the previous best bounds were O(d) on the nilpotence, and an ineffective function of d on the index. In fact, we obtain this as a small part of a much more detailed fine-scale description of the structure of G. These results have a wide range of applications in various aspects of the theory of vertex-transitive graphs: percolation theory, random walks, structure of finite groups, scaling limits of finite vertex-transitive graphs. After an overview of these applications, I will try to present some of the key concepts behind the proof.

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