On ampleness and pseudo-Anosov homeomorphisms in the free group

TL;DR

This paper demonstrates that the first order theory of non-abelian free groups is n-ample for any n, using pseudo-Anosov homeomorphisms, and provides alternative proofs related to imaginaries in hyperbolic groups.

Contribution

It introduces a novel approach using pseudo-Anosov homeomorphisms to prove ampleness in free groups and offers alternative proofs to existing results on imaginaries.

Findings

The first order theory of free groups is n-ample for all n.

Provides an alternative proof to Ould Houcine-Tent's main result.

Offers new insights into imaginaries in hyperbolic groups.

Abstract

We use pseudo-Anosov homeomorphisms of surfaces in order to prove that the first order theory of non abelian free groups, $T_{fg}$, is $n$-ample for any $n\in\omega$. This result adds to the work of Pillay, that proved that $T_{fg}$ is non CM -trivial. The sequence witnessing ampleness is a sequence of primitive elements in $F_{\omega}$. Our result provides an alternative proof to the main result of a preprint by Ould Houcine-Tent. We also add an appendix in which we make a few remarks on Sela's paper on imaginaries in torsion free hyperbolic groups. In particular we give alternative transparent proofs concerning the non-elimination of certain imaginaries.