Infinitesimal containment and sparse factors of iid

Abstract

We introduce infinitesimal weak containment for measure-preserving actions of a countable group $\Gamma$: an action $(X,\mu)$ is infinitesimally contained in $(Y,\nu)$ if the statistics of the action of $\Gamma$ on small measure subsets of $X$ can be approximated inside $Y$. We show that the Bernoulli shift $[0,1]^\Gamma$ is infinitesimally contained in the left-regular action of $\Gamma$. For exact groups, this implies that sparse factor-of-iid subsets of $\Gamma$ are approximately hyperfinite. We use it to quantify a theorem of Chifan--Ioana on measured subrelations of the Bernoulli shift of an exact group. For the proof of infinitesimal containment we define \emph{entropy support maps}, which take a small subset $U$ of $\{0,1\}^I$ and assign weights to coordinates above every point of $U$, according to how ''important'' they are for the structure of the set.