Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On piecewise hyperdefinable groups

Arturo Rodriguez Fanlo

TL;DR

The paper extends Hrushovski’s stable-group theory framework to piecewise hyperdefinable groups, establishing a robust structure for Lie cores and a stabilizer theory in this broader setting. It develops a comprehensive theory of piecewise hyperdefinable sets, including their logic topologies, spaces of types, and metrisation, and then builds Lie-core machinery to analyze piecewise hyperdefinable groups via canonical components such as $G^{000}_A$, $G^{00}_A$, $G^0_A$, and the aperiodic part $G^{ ext{ap}}$, culminating in a minimal Lie core representation $L$ and a global Lie-core map. The Stabilizer Theorem is generalized to this context, with a refined analysis of dividing, forking, and stable relations, yielding a wide, normal stabilizer under natural hypotheses and eliminating the need for right-translation invariance in the core results. A Rough Lie Model Theorem is developed for near-subgroups, showing how piecewise hyperdefinable groups with near-subgroups admit connected Lie models with explicit control over the kernel and domain, preserving parameter-independence aspects. Overall, the work provides foundational tools to study rough approximate subgroups and their Lie-model structures in metric and topological-group settings using a model-theoretic, language-expansion-insensitive approach.

Abstract

The aim of this paper is to generalize and improve two of the main model-theoretic results of "Stable group theory and approximate subgroups" by E. Hrushovski to the context of piecewise hyperdefinable sets. The first one is the existence of Lie models. The second one is the stabilizer theorem. In the process, a systematic study of the structure of piecewise hyperdefinable sets is developed. In particular, we show the most significant properties of their logic topologies.

On piecewise hyperdefinable groups

TL;DR

The paper extends Hrushovski’s stable-group theory framework to piecewise hyperdefinable groups, establishing a robust structure for Lie cores and a stabilizer theory in this broader setting. It develops a comprehensive theory of piecewise hyperdefinable sets, including their logic topologies, spaces of types, and metrisation, and then builds Lie-core machinery to analyze piecewise hyperdefinable groups via canonical components such as , , , and the aperiodic part , culminating in a minimal Lie core representation and a global Lie-core map. The Stabilizer Theorem is generalized to this context, with a refined analysis of dividing, forking, and stable relations, yielding a wide, normal stabilizer under natural hypotheses and eliminating the need for right-translation invariance in the core results. A Rough Lie Model Theorem is developed for near-subgroups, showing how piecewise hyperdefinable groups with near-subgroups admit connected Lie models with explicit control over the kernel and domain, preserving parameter-independence aspects. Overall, the work provides foundational tools to study rough approximate subgroups and their Lie-model structures in metric and topological-group settings using a model-theoretic, language-expansion-insensitive approach.

Abstract

The aim of this paper is to generalize and improve two of the main model-theoretic results of "Stable group theory and approximate subgroups" by E. Hrushovski to the context of piecewise hyperdefinable sets. The first one is the existence of Lie models. The second one is the stabilizer theorem. In the process, a systematic study of the structure of piecewise hyperdefinable sets is developed. In particular, we show the most significant properties of their logic topologies.

Paper Structure

This paper contains 20 sections, 98 theorems, 21 equations.

Table of Contents

  1. Piecewise hyperdefinable sets
  2. Hyperdefinable sets
  3. The logic topologies of hyperdefinable sets
  4. Piecewise hyperdefinable sets
  5. The logic topologies of piecewise hyperdefinable sets
  6. Spaces of types
  7. Metrisation results
  8. Examples
  9. Piecewise hyperdefinable groups
  10. Preliminaries on topological groups
  11. Local compactness and generic pieces
  12. Model-theoretic components
  13. Lie cores
  14. Digression on Breuillard-Green-Tao Classification
  15. Stabilizer Theorem
  16. ...and 5 more sections

Key Result

Lemma 1.1

Let $P=\hbox{$X$}/\hbox{$E$}$ be an $A^*$-hyperdefinable set. Then, the image by $\mathrm{quot}_P$ of any $\bigwedge_{A^*}$-definable subset of $X$ is an $\bigwedge_{A^*}$-definable subset of $P$. Moreover, the preimage function $\mathrm{Im}^{-1}\mathrm{quot}_P$ is an isomorphism, whose inverse is $

Theorems & Definitions (214)

  • Lemma 1.1: Correspondence Lemma
  • Example 1.2
  • Lemma 1.3
  • proof
  • Remark 1.4
  • Lemma 1.5
  • proof
  • Corollary 1.6
  • proof
  • Corollary 1.7
  • ...and 204 more