The theory of locally compact groups stretches out between two antipodes: on one hand, connected groups whose structure, according to the solution to Hilbert fifth problem, is governed by Lie theory and is thus relatively rigid, and on the other hand, discrete groups, which are subject to a spectacular variety of behaviours, going from the most stringent rigidity properties to the most intriguing pathological ones. The goal of this research program is to explore the wide space lying between these two extremes.
The entire program is built around two major open problems: performing an exhaustive study of compactly generated simple locally compact groups, and finding an algebraic characterization of those locally compact groups which are linear. Although these problems do not seem to be directly approachable given the current state of knowledge, they are nevertheless considered as guidelines suggesting a number of specific questions and conjectures which are envisaged in detail under various perspectives of algebraic, geometric, arithmetic and analytic nature. Each of these specific questions presents independent interest; answers to any of them will moreover provide insight into the guiding problems.
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