Left-preorder spaces of free products have no isolated elements; when the factors are finitely generated, every nonempty such space is a Cantor set.
Free products of bi-orderable groups
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We prove a bi-ordered version of Rivas' result for free products of left-order groups. Namely, we show that a free product of bi-ordered groups does not admit isolated bi-ordering. Our method relies on the dynamical realization of bi-ordered groups. We also show that the natural action of the automorphism group $Aut(F_2)$ on $F_2$ does not have dense orbits.
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Spaces of left-preorders on free products
Left-preorder spaces of free products have no isolated elements; when the factors are finitely generated, every nonempty such space is a Cantor set.