Measure growth in compact semisimple Lie groups and the Kemperman Inverse Problem

Abstract

Suppose $G$ is a compact semisimple Lie group, $\mu$ is the normalized Haar measure on $G$, and $A,A^2\subseteq G$ are measurable. We show that $$\mu (A^2)\ge \min \{1,2\mu (A)+\eta \mu (A)(1-2\mu (A))\}$$ with the absolute constant $\eta >0$ (independent from the choice of $G$) quantitatively determined. We also show a more general result for connected compact groups without a toric quotient and resolve the Kemperman Inverse Problem from 1964.

50 pages; the paper subsumes the compact case of arxiv:2006.01824. The proof now is simplified and effective, the smallness assumption in the main theorem is removed, and the model theoretical arguments are no longer needed. For the non-compact case, a more straightforward proof using the non-abelian Brunn-Minkowski inequality was obtained by An, Zhang, and the authors in arXiv:2111.05236