Infinitesimal containment and sparse factors of iid
Mikołaj Frączyk
TL;DR
The paper develops a new notion of infinitesimal containment for measure-preserving actions of countable groups, capturing how statistics of tiny subsets reflect within another action. It proves that the Bernoulli shift is infinitesimally contained in the group’s left-regular action, and, for exact groups, that sparse factor-of-iid subsets are approximately hyperfinite, connecting local subset dynamics to global orbit-structure properties. A central technical advance is the entropy- and conformal-σ-algebra framework, via the entropy-support maps, which yields quantitative containment results and enables a strengthened Chifan–Ioana theorem for subrelations. The work also introduces unimodular random subsets and thinnings as robust machinery to study limits and hyperfiniteness, and it outlines several open questions and potential extensions to compact actions. Overall, the results provide a new toolkit linking infinitesimal dynamics, entropy, and measured group theory with implications for cost, fixed price, and the geometry of sparse random subsets.
Abstract
We introduce infinitesimal weak containment for measure-preserving actions of a countable group $Γ$: an action $(X,μ)$ is infinitesimally contained in $(Y,ν)$ if the statistics of the action of $Γ$ on small measure subsets of $X$ can be approximated inside $Y$. We show that the Bernoulli shift $[0,1]^Γ$ is infinitesimally contained in the left-regular action of $Γ$. For exact groups, this implies that sparse factor-of-iid subsets of $Γ$ 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.