Measure expanding actions, expanders and warped cones

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.

33 pages, made a number of improvements and corrections throughout. To appear in Transactions of the AMS

References in corpus (3)

• Measurable equidecompositions for group actions with an expansion property
• Warped cones and spectral gaps
• Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts

Cited by in corpus (10)

• Superexpanders from group actions on compact manifolds
• Warped cones, (non-)rigidity, and piecewise properties, with a joint appendix with Dawid Kielak
• Straightening warped cones
• Nonpositive curvature is not coarsely universal
• Sum-product for real Lie groups
• Warped cones and proper affine isometric actions of discrete groups on Banach spaces
• A Markovian and Roe-algebraic approach to asymptotic expansion in measure
• Rigidity of warped cones and coarse geometry of expanders
• Asymptotic expansion in measure and strong ergodicity
• On the structure of asymptotic expanders