JSJ decompositions of doubles of free groups

TL;DR

This paper classifies all JSJ decompositions of doubles of free groups of rank two and analyzes their Makanin-Razborov diagrams, revealing that limit groups are generally not freely subgroup separable.

Contribution

It provides a complete classification of JSJ decompositions for doubles of free groups of rank two and explores their implications on subgroup separability.

Findings

Classified all JSJ decompositions of doubles of free groups of rank two.

Computed the Makanin-Razborov diagram for a specific double.

Showed that limit groups are not generally freely subgroup separable.

Abstract

We classify all possible JSJ decompositions of doubles of free groups of rank two and we then compute the Makanin-Razborov diagram of a particular double of a free group and deduce that in general limit groups are not freely subgroup separable.