Generalized locally compact models for approximate groups

Krzysztof KrupiÅski; Anand Pillay

mathematical logic

Generalized locally compact models for approximate groups

arXiv:2310.20683

summary

The paper gives a simpler proof that every approximate subgroup has a generalized definable locally compact model, using basic model‑theoretic tools and Ellis groups from topological dynamics, and shows this model is universal in an appropriate category.

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 $SL_{2} (R)$ .

Topics & keywords

#approximate groups #definable groups #topological dynamics #locally compact models #Ellis groups #generic subsetsapproximate subgroupdefinable locally compact modelEllis groupmodel theorytopological dynamicsuniversal cover of SL2(R)