Suspensions of Bernoulli shifts

TL;DR

This paper demonstrates that Bernoulli shifts over finitely generated groups can be embedded into spaces of pointed trees, enabling a suspension construction and establishing universality for certain transformation pseudogroups.

Contribution

It introduces a method to embed Bernoulli shifts into spaces of pointed trees and constructs their suspension, connecting group actions with foliated spaces.

Findings

Bernoulli shifts can be embedded into pointed trees spaces

The space of pointed trees is universal for expansive pseudogroups

Suspensions of Bernoulli shifts can be realized via foliated spaces

Abstract

We show that for a given finitely generated group, its Bernoulli shift space can be equivariantly embedded as a subset of a space of pointed trees with Gromov-Hausdorff metric and natural partial action of a free group. Since the latter can be realised as a transverse space of a foliated space with leaves Riemannian manifolds, this embedding allows us to obtain a suspension of such Bernoulli shift. By a similar argument we show that the space of pointed trees is universal for compactly generated expansive pseudogroups of transformations.