On random walks in the group of Euclidean isometries

In contrast to the two dimensional case, in dimension $d \geq 3$ averaging operators on the $d-1$-sphere using finitely many rotations, i.e. averaging operators of the form $Af(x)= |S|^{-1} \sum_{\theta \in S} f(s x)$ where $S$ is a finite subset of $\mathrm{SO}(d)$, can have a spectral gap on $L^2$ of the $d - 1$-sphere. A result of Bourgain and Gamburd shows that this holds, for instance, for any finite set of elements in $\mathrm{SO}(3)$ with algebraic entries and spanning a dense subgroup. We prove a new spectral gap result for averaging operators corresponding to finite subsets of the isometry group of $\mathbb R^d$, which is a semi-direct product of $\mathrm{SO}(d)$ and $\mathbb R^d$, provided the averaging operator corresponding to the rotation part of these elements have a spectral gap. This new spectral gap result has several applications, and in particular (sharpening a previous result by Varju) allows us to prove a local-central limit theorem for a random walks on $\mathbb R^d$ using the elements of the isometry group that holds up to an exponentially small scale, as well as to the study of self similar measures in $d \geq 3$ dimensions. Time permitting we also present a new family of expanders that can be constructed using similar tools. This talk is based on joint work with P. Varju.