Measure expanding actions, expanders and warped cones

TL;DR

This paper introduces a method to approximate measure space actions with finite graphs, demonstrating their expander properties are equivalent to the actions being expanding in measure, and relates these graphs to warped cones for embedding results.

Contribution

It provides a unified framework linking measure actions, expanders, and warped cones, and establishes new non-embeddability results and connections to existing conjectures.

Findings

Graphs form expanders iff actions are expanding in measure

Graphs are quasi-isometric to warped cone level sets

Proves non-embeddability of warped cones and relates to a conjecture

Abstract

We define a way of approximating actions on measure spaces using finite graphs; we then show that in quite general settings these graphs form a family of expanders if and only if the action is expanding in measure. This provides a somewhat unified approach to construct expanders. We also show that the graphs we obtain are uniformly quasi-isometric to the level sets of warped cones. This way we can also prove non-embeddability results for the latter and restate an old conjecture of Gamburd-Jakobson-Sarnak.