Let $SO(3,R)$ be the $3D$-rotation group equipped with the real-manifold topology and the normalized Haar measure $\mu$. Confirming a conjecture by Breuillard and Green, we show that if $A$ is an open subset of $SO(3,R)$ with sufficiently small measure, then $\mu(A^2) > 3.99 \mu(A)$. This is joint work with Chieu-Minh Tran (NUS) and Ruixiang Zhang (Berkeley). 

In a recent work with Sam Mellick and Amanda Wilkens, we proved that higher rank semisimple Lie groups satisfy a generalization of Gaboriau fixed price property (originally defined for countable groups) to the setting of locally compact second countable groups. As one of the corollaries, under mild conditions, we can prove that the rank (minimal number of generators) or the first mod-p Betti number of a higher rank lattice grow sublinearly in the covolume.  The proof relies on surprising geometric properties of Poisson-Voronoi tessellations in higher-rank symmetric spaces, which could be of independent interest.