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On the model theory of the Free Factor Complex of rank 2

Zahra Mohammadi Khangheshlaghi, Katrin Tent

Abstract

We begin the investigation of the free factor complex of a free group of finite rank. For the case of rank 2 we axiomatize its theory and show that it is $ω$-stable with prime model $AF_2$.

On the model theory of the Free Factor Complex of rank 2

Abstract

We begin the investigation of the free factor complex of a free group of finite rank. For the case of rank 2 we axiomatize its theory and show that it is -stable with prime model .
Paper Structure (10 sections, 29 theorems, 21 equations)

This paper contains 10 sections, 29 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. The Free Factor Complex of rank 2
  3. An additional graph structure on $\mathcal{AF}_2$
  4. Paths avoiding conjugacy classes
  5. Some bounds on blocks
  6. The Theory of the Free Factor Complex of rank 2
  7. Stability
  8. Algebraic closure
  9. From models of $Th(\mathcal{AF}_2)$ to models of $Th(\mathcal{OF}_2)$
  10. Appendix

Key Result

Lemma 3.3

If $x, y, z \in \mathcal{AF}_2$ are distinct conjugates, then

Theorems & Definitions (87)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Definition 3.2
  • Lemma 3.3
  • ...and 77 more