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Warped cone is a geometric object associated with a measure preserving isometric action of a finitely generated group on a compact manifold. It encodes the geometry of the manifold, geometry of the group (Cayley graph) and the dynamics of the group. This geometric object has been introduced by J. Roe in the context of Coarse Baum-Connes conjecture (CBC conjecture). Warped cones associated with the action of amenable groups give examples of CBC conjecture and some expander graphs can be constructed from the warped cones associated with the action of Property (T) group. On the other hand, Measured Equivalence (ME) is an equivalence relation between two countable groups introduced by M. Gromov as a measure-theoretic analogue of quasi-isometry. If the `cocyles' associated with a measured equivalence relation are bounded, the relation is called Uniform Measured Equivalence. In this lecture, we prove that if two warped cones are quasi-isometric, then the associated groups are Uniform Measured Equivalent. As an application, we will talk about different ME-invariants which distinguish two warped cones up to quasi-isometry. This is a work in progress.
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