Asymptotic expansion in measure and strong ergodicity

Abstract

In this paper, we introduce and study a notion of asymptotic expansion in measure for measurable actions. This generalises expansion in measure and provides a new perspective on the classical notion of strong ergodicity. Moreover, we obtain structure theorems for asymptotically expanding actions, showing that they admit exhaustions by domains of expansion. As an application, we recover a recent result of Marrakchi, characterising strong ergodicity in terms of local spectral gaps. We also show that homogeneous strongly ergodic actions are always expanding in measure and establish a connection between asymptotic expansion in measure and asymptotic expanders by means of approximating spaces.

arXiv commentTo appear in Journal of Topology and Analysis

References in corpus (5)

• Measure expanding actions, expanders and warped cones
• Measurable equidecompositions for group actions with an expansion property
• Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts
• Quasi-local Algebras and Asymptotic Expanders
• A Markovian and Roe-algebraic approach to asymptotic expansion in measure

Cited by in corpus (1)

• A Markovian and Roe-algebraic approach to asymptotic expansion in measure