Submission date: 31 October 2023

Abstract:

We give a proof of the existence of generalized definable locally compact models for arbitrary approximate subgroups via an application of topological dynamics in model theory. Our construction is simpler and shorter than the original one obtained by Hrushovski in Beyond the Lascar group, and it uses only basic model theory (mostly spaces of types and realizations of types). The main tools are Ellis groups from topological dynamics considered for suitable spaces of types. However, we need to redevelop some basic theory of topological dynamics for suitable locally compact flows in place of (compact) flows. We also prove that the generalized definable locally compact model which we constructed is universal in an appropriate category. We note that the main result yields structural information on definable generic subsets of definable groups, with a more precise structural result for generics in the universal cover of SL_2(ℝ).

Mathematics Subject Classification: 03C60, 03C98, 37B02, 54H11, 11B30, 11P70, 20A15, 20N99

Keywords and phrases:

Full text arXiv 2310.20683: pdf, ps.