On groups definable in $p$-adically closed fields

This paper is about the $dfg$/$fsg$ decomposition for groups $G$ definable in $p$-adically closed fields. It is proved that for $G$ definably amenable, $G$ has a definable normal $dfg$ subgroup $H$ such that the quotient $G/H$ is a definable $fsg$ group. The result was known for groups definable in $o$-minimal expansions of real closed fields (see \cite{C-P-o-mini}). We also give a version for arbitrary (not necessarily definably amenable) groups $G$ definable in $p$-adically closed fields: there is a definable $dfg$ subgroup $H$ of $G$ such that the homogeneous space $G/H$ is definable and definably compact. (In the $o$-minimal case this is Fact 3.25 of \cite{Peterzil-Starchenko-mutypes}). Note that $dfg$ stands for has a definable $f$-generic type", and $fsg$ for has finitely satisfiable generics", which will be discussed together with various equivalences. We will need to understand something about groups of the form $G(k)$ where $k$ is a $p$-adically closed field and $G$ a semisimple algebraic group over $k$, and as part of the analysis we will prove the Kneser-Tits conjecture over $p$-adically closed fields.

💡 Research Summary

The paper investigates groups definable in p‑adically closed fields (the theory pCF) and establishes a structural decomposition analogous to the dfg/fsg decomposition known for groups definable in o‑minimal expansions of real closed fields. The authors work in a monster model of pCF and assume familiarity with Keisler measures, NIP, and distal theories.

Key concepts.

• Definably amenable: a group admits a left‑invariant Keisler measure on definable subsets.
• dfg (definable f‑generic): there exists a global type p concentrating on the group such that every left translate of p is definable over a small model.
• fsg (finitely satisfiable generics): a global type p whose every left translate is finitely satisfiable in a fixed small model.
• Definable compactness: for definable subsets of a p‑adic field, this coincides with being closed and bounded in the usual valuation topology.

Main results.

1. Theorem A (Theorem 8.8). If G is a definably amenable group definable in a p‑adic field M, then there exists a definable normal subgroup H ⊲ G which is dfg, and the quotient C = G/H is a definable fsg group. Thus G admits a dfg/fsg short exact sequence 1 → H → G → C → 1.
2. Theorem B (Theorem 6.4, 8.17). For an arbitrary definable group G (no amenability assumption) there is a definable dfg subgroup H such that the homogeneous space G/H is definably compact; consequently G/H is fsg.

These theorems answer positively Question 1.19 from