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Preprint Number 1682

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1682. Anand Pillay and Ningyuan Yao
Definable f-Generic Groups over p-Adic Numbers

Submission date: 4 November 2019


arXiv admin note: text overlap with arXiv:1901.09508

The aim of this paper is to develop the theory for definable f-generic groups in the p-adic field within the framework of definable topological dynamics, here the “definable f-generic” means a definable group admits a global f-generic type which is definable over a small submodel. This “definable f-generic” is a dual concept to “finitely satisfiable generic”, and a useful tool to describe the analogue of torsion free o-minimal groups in the p-adic context.
In this paper we will show that every definable f-generic group definable in ℚ_p is eventually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ℚ_p. This is analogous to the o-minimal context, where every connected torsion free group definable in ℝ is isomorphic to a trigonalizable algebraic group (Lemma 3.4, [COS]). We will also show that every open definable f-generic subgroup of a definable f-generic group has finite index, and every f-generic type of a definable f-generic group is almost periodic, which gives a positive answer on the problem raised in [P-Y] of whether f-generic types coincide with almost periodic types in the p-adic case.

Mathematics Subject Classification: 03C98, 20G25, 22E35

Keywords and phrases:

Full text arXiv 1911.01833: pdf, ps.

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