Dynamics seminar: Hagai Lavner (HUJI) Random mapping statistics, growth of unbounded subsets of nilpotent groups, and geometry of positive group laws

Tue, 21/06/202214:00-15:00

The growth of |S^n| for a fixed finite subset S of a group G as n\to\infty is well studied.
In this talk we will be concerned with the maximum size of S^n over all n-element subsets S.
A result of Shalev and Semple shows that when G is f.g. residually finite, this is n^n unless G is virtually nilpotent.
Our main result quantifies this by showing that unless G is virtually abelian, there is a subset of size n with |S^n| at least (e^(-1/4)+o(1))n^n (which is tight for the Heisenberg group).
We shall discuss the proof, which proceeds via inquiring when one can retrieve a function f:[n]\to[n] from a certain pair-statistic,
and also describe the geometry of (positive) group laws.

This is joint work with Be'eri Greenfeld.

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