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First-order sentences in random groups I: universal sentences

O. Kharlampovich, R. Sklinos

Abstract

We prove that a random group, in Gromov's density model with $d<1/16$, satisfies a universal sentence $σ$ (in the language of groups) if and only if $σ$ is true in a nonabelian free group.

First-order sentences in random groups I: universal sentences

Abstract

We prove that a random group, in Gromov's density model with , satisfies a universal sentence (in the language of groups) if and only if is true in a nonabelian free group.

Paper Structure

This paper contains 18 sections, 47 theorems, 34 equations, 30 figures.

Table of Contents

  1. Introduction
  2. First-order Logic & Axioms for the Universal Theory of the Free Group
  3. Models of randomness and background results
  4. The models
  5. A random group is not a limit group
  6. Van Kampen diagrams
  7. Reducing universal sentences to equations in random groups
  8. Bounding the length of short solutions
  9. Actions on real trees
  10. Diagrams in groups satisfying small cancellation properties
  11. Tripod stabilizers
  12. Segment stabilizers
  13. JSJ decompositions and approximations
  14. Short solutions
  15. Diagrams for solutions of equations
  16. ...and 3 more sections

Key Result

proposition 1

Rem A finitely generated group $G$ is a model of $T_{fg}^{\forall}$ if and only if it is fully residually free.

Figures (30)

  • Figure 1: Non-filamentous van Kampen Diagrams
  • Figure 2: A filamentous topological van Kampen Diagram
  • Figure 3: A geodesic triangle in a Cayley graph.
  • Figure 4: An arbitrary tessellation of a digon.
  • Figure 5: Simple geodesic digon and triangles
  • ...and 25 more figures

Theorems & Definitions (101)

  • definition 1
  • definition 2
  • proposition 1
  • corollary 1
  • proof
  • remark 1
  • proposition 2
  • proof
  • lemma 1
  • proof
  • ...and 91 more