On groups definable in $p$-adically closed fields

TL;DR

This work extends the $dfg$/$fsg$ decomposition to groups definable in $p$-adically closed fields, proving that definably amenable $G$ admit a definable normal $dfg$ subgroup $H$ with $G/H$ definable $fsg$, and, more generally, that $G$ has a definable $dfg$ subgroup with a definably compact quotient. It then establishes a $p$-adic analogue of the Kneser–Tits conjecture, showing that open subgroups of definably simple groups are either definably compact or of finite index, and that abstract simplicity follows in the non-compact, definably simple case. The paper develops a robust framework for $dfg$ groups and their definable families via pro-definability, uniform definability of types, and definable compactness, leading to a complete $dfg$/$fsg$ decomposition and a theory of definably amenable components. These results address a distal setting analogue of Peterzil–Steinhorn type decompositions in the real case and yield applications to weak generics, almost periodic types, and the structure of definably amenable components, with explicit consequences for groups definable over $Q_p$. Overall, the work provides tools for understanding definable groups in $p$-adic settings and confirms a positive answer to the $dfg$/$fsg$ decomposition in $p$CF, including new insights into Kneser–Tits and definable amenability.

Abstract

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.

Theorems & Definitions (111)

• Corollary 1.1
• Definition 2.1
• Remark 2.11
• Remark 2.13
• Corollary 2.15
• proof
• Claim 2.19
• proof
• Claim 2.20
• proof