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One-dimensional subgroups and connected components in non-abelian $p$-adic definable groups

Will Johnson, Ningyuan Yao

Abstract

We generalize two of our previous results on abelian definable groups in $p$-adically closed fields to the non-abelian case. First, we show that if $G$ is a definable group that is not definably compact, then $G$ has a one-dimensional definable subgroup which is not definably compact. This is a $p$-adic analogue of the Peterzil-Steinhorn theorem for o-minimal theories. Second, we show that if $G$ is a group definable over the standard model $\mathbb{Q}_p$, then $G^0 = G^{00}$. As an application, definably amenable groups over $\mathbb{Q}_p$ are open subgroups of algebraic groups, up to finite factors. We also prove that $G^0 = G^{00}$ when $G$ is a definable subgroup of a linear algebraic group, over any model.

One-dimensional subgroups and connected components in non-abelian $p$-adic definable groups

Abstract

We generalize two of our previous results on abelian definable groups in -adically closed fields to the non-abelian case. First, we show that if is a definable group that is not definably compact, then has a one-dimensional definable subgroup which is not definably compact. This is a -adic analogue of the Peterzil-Steinhorn theorem for o-minimal theories. Second, we show that if is a group definable over the standard model , then . As an application, definably amenable groups over are open subgroups of algebraic groups, up to finite factors. We also prove that when is a definable subgroup of a linear algebraic group, over any model.
Paper Structure (15 sections, 35 theorems, 24 equations)

This paper contains 15 sections, 35 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Outline
  4. Preliminaries
  5. Definable Compactness
  6. $p\text{CF}$ and definable groups
  7. Algebraic Groups
  8. Centralizer-connected groups
  9. "Affine" groups and the adjoint action
  10. Definable compactness in extensions and quotients
  11. The $p$-adic Peterzil-Steinhorn theorem
  12. Reduction to the standard model $\mathbb{Q}_p$
  13. The case of $\mathbb{Q}_p$
  14. Compact Hausdorff groups and $G/G^{00}$
  15. $G^0$ vs $G^{00}$

Key Result

Theorem 1.1

Let $G$ be a definable group in a $p$-adically closed field. If $G$ is not definably compact, then $G$ contains a one-dimensional definable subgroup $H$ which is not definably compact.

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 2.2
  • Definition 2.4
  • Theorem 2.5
  • proof
  • ...and 67 more