Submission date: 24 January 2009.

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 experimental- 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.

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