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Generic differentiability and $P$-minimal groups

Will Johnson

TL;DR

This work extends the Pillay program to $P$-minimal theories by establishing generic differentiability for definable functions and using it to prove an analogue of Pillay’s conjecture for definable groups: every $n$-dimensional definable group in a highly saturated $P$-minimal structure has an open definable subgroup $H$ that is compactly dominated by $H^{00}$ and whose quotient $H/H^{00}$ is a $p$-adic Lie group of the expected dimension. The approach hinges on definable compactness and a detailed analysis of asymptotic behavior near 0, yielding an asymptotic $f(a+x)-f(a)\nobreak o C(a)x^q$ with $q=1$, plus an inverse-function theorem. The paper also classifies interpretable fields in $P$-minimal structures using the generic differentiability framework, and provides two general proofs of the $P$-minimal Onshuus–Pillay conjecture, plus a third proof in the pure theory of $p$-adically closed fields. Collectively, these results connect $P$-minimal groups with $p$-adic Lie groups and yield a robust model-theoretic analogue of the o-minimal Pillay conjectures, with broad implications for the structure theory of definable groups in $p$-adic settings.

Abstract

We prove generic differentiability in $P$-minimal theories, strengthening an earlier result of Kuijpers and Leenknegt. Using this, we prove Onshuus and Pillay's $P$-minimal analogue of Pillay's conjectures on o-minimal groups. Specifically, let $G$ be an $n$-dimensional definable group in a highly saturated model $M$ of a $P$-minimal theory. Then there is an open definable subgroup $H \subseteq G$ such that $H$ is compactly dominated by $H/H^{00}$, and $H/H^{00}$ is a $p$-adic Lie group of the expected dimension. Additionally, the generic differentiability theorem immediately implies a classification of interpretable fields in $P$-minimal theories, by work of Halevi, Hasson, and Peterzil.

Generic differentiability and $P$-minimal groups

TL;DR

This work extends the Pillay program to -minimal theories by establishing generic differentiability for definable functions and using it to prove an analogue of Pillay’s conjecture for definable groups: every -dimensional definable group in a highly saturated -minimal structure has an open definable subgroup that is compactly dominated by and whose quotient is a -adic Lie group of the expected dimension. The approach hinges on definable compactness and a detailed analysis of asymptotic behavior near 0, yielding an asymptotic with , plus an inverse-function theorem. The paper also classifies interpretable fields in -minimal structures using the generic differentiability framework, and provides two general proofs of the -minimal Onshuus–Pillay conjecture, plus a third proof in the pure theory of -adically closed fields. Collectively, these results connect -minimal groups with -adic Lie groups and yield a robust model-theoretic analogue of the o-minimal Pillay conjectures, with broad implications for the structure theory of definable groups in -adic settings.

Abstract

We prove generic differentiability in -minimal theories, strengthening an earlier result of Kuijpers and Leenknegt. Using this, we prove Onshuus and Pillay's -minimal analogue of Pillay's conjectures on o-minimal groups. Specifically, let be an -dimensional definable group in a highly saturated model of a -minimal theory. Then there is an open definable subgroup such that is compactly dominated by , and is a -adic Lie group of the expected dimension. Additionally, the generic differentiability theorem immediately implies a classification of interpretable fields in -minimal theories, by work of Halevi, Hasson, and Peterzil.
Paper Structure (26 sections, 53 theorems, 143 equations)

This paper contains 26 sections, 53 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Variants of Theorem \ref{['intro-1']}
  4. The proof of generic differentiability
  5. Conventions
  6. Definable compactness
  7. Applications of definable compactness
  8. More tools
  9. Chains of nowhere dense sets
  10. Strengthening limits
  11. Rational powers
  12. Asymptotic behavior of functions in one variable
  13. Generic differentiability
  14. Review of strict differentiability
  15. Generic (strict) differentiability in one variable
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $M$ be a model of a $P$-minimal theory $T$. Let $U \subseteq M^n$ be a non-empty definable open set. Let $f : U \to M^m$ be a definable function. Then there is a definable open set $U_0 \subseteq U$ such that $\dim(U \setminus U_0) < n$ and $f$ is differentiable on $U_0$.

Theorems & Definitions (133)

  • Theorem 1.1
  • Corollary 1.2: hhp
  • Conjecture 1.3: Onshuus-Pillay
  • Theorem 1.4: Theorem \ref{['second-main']}
  • Conjecture 1.5: Modified Onshuus-Pillay conjecture
  • Theorem 1.6: Theorem \ref{['third-main']}
  • Theorem 1.7: Theorem \ref{['last-main']}
  • Lemma 2.2
  • proof
  • Remark 2.4
  • ...and 123 more