Stability and Bounded Balls of Free Products

TL;DR

This paper explores the stability properties of free products of stable groups, providing preliminary results on quantifier-free definable sets and bounded balls, contributing to the understanding of their model-theoretic behavior.

Contribution

It offers initial, experimental insights into the stability of free products of stable groups, focusing on quantifier-free definable sets and bounded balls.

Findings

Preliminary evidence suggests stability in free products under certain conditions.

Focus on quantifier-free definable sets and bounded balls.

Results are experimental and suggest directions for future research.

Abstract

In a series of papers starting in [Sel01] and culminating in [Sel07], Z. Sela proved that free groups, and more generally torsion-free hyperbolic groups, have a stable first-order theory. The question of the stability of the free product of two arbitrary stable groups has then been raised by E. Jaligot with, seemingly, the reasonable conjecture of a positive answer [Jal08]. The complete proof however will be a grand project of generalization of above papers of Sela.In the meantime, we provide here a very preliminary -or somehow experi- mental- result in the direction of the stability of free products of stable groups, restricting ourselves to quantifer-free definable sets and to bounded balls of free products.