UB12 · Ergodic Group Theory

The aim of the proposed research is gaining a better understanding of locally compact groups and their lattices. Our tools are mainly ergodic theoretical.We propose a variety of novel ideas that open new horizons for research.The first meta idea is the adoption of tools from the semi-simple theory in order to apply them for general locally compact groups. In particular, we suggest a construction of a ""Weyl group"" and an abstract definition of rank for every locally compact group. We are able to construct a ""Coxeter complex"" and we foresee a construction of a ""building like"" object.A second set of ideas concerns the category of measure equivalences, which is a natural generalization of the notion of a lattice in a group. This category is long known to be a measurable counterpart of the better studied category of quasi-isometries, yet it misses a good definition of self measure equivalences of an object, analog to the group of quasi-isometries.We suggest such a definition, and propose to study it, among a variety of related constructions.A full implementation of our ideas requires a better understanding of locally compact groups.Thus, an important aspect of the proposed research is that it leaves plenty of room for the study of specific examples and test cases.""