Infinitesimal containment and sparse factors of iid

TL;DR

The paper introduces infinitesimal weak containment for measure-preserving group actions, showing Bernoulli shifts are contained in regular actions, and explores implications for sparse factors and hyperfiniteness in exact groups.

Contribution

It defines infinitesimal weak containment, proves Bernoulli shifts are contained in regular actions, and applies this to analyze sparse factors and subrelations in exact groups.

Findings

Bernoulli shift is infinitesimally contained in the regular action

Sparse factor-of-iid subsets are approximately hyperfinite for exact groups

Quantifies Chifan--Ioana's theorem on subrelations of Bernoulli shifts

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.