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Fields interpretable in the free group

Rizos Sklinos

Abstract

We prove that no infinite field is interpretable in the first-order theory of nonabelian free groups. We also obtain a characterization of Abelian groups interpretable in this theory.

Fields interpretable in the free group

Abstract

We prove that no infinite field is interpretable in the first-order theory of nonabelian free groups. We also obtain a characterization of Abelian groups interpretable in this theory.

Paper Structure

This paper contains 28 sections, 33 theorems, 42 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Overview
  3. Strategy:
  4. Paper structure:
  5. Some model theory
  6. Imaginaries
  7. Internality & One-Basedness
  8. Stars of groups
  9. Reduced Forms
  10. Permuting the rays
  11. Towers
  12. The construction of a tower
  13. Multiplets of towers
  14. Closures of towers
  15. Symmetrising closures of multiplets of towers
  16. ...and 13 more sections

Key Result

theorem 1

Let $\F$ be a nonabelian free group. Let $E(\bar{x},\bar{y})$ be a definable equivalence relation in $\F$, with $\abs{\bar{x}}=m$. Then there exist $k,\ell<\omega$ and a definable relation: such that:

Figures (7)

  • Figure 1: A star of groups
  • Figure 2: The characteristic set of $h$ in the case $h$ is hyperbolic it does not contain $*$.
  • Figure 3: A centered splitting
  • Figure 4: An Abelian floor
  • Figure 5: A tower of height 4.
  • ...and 2 more figures

Theorems & Definitions (100)

  • proof
  • definition 1
  • definition 2
  • remark 1
  • theorem 1
  • remark 2
  • definition 3
  • lemma 1
  • proof
  • definition 4
  • ...and 90 more