Are there any references how to show that:the free products of finitely many finitely generated groups are hyperbolic relative to the free factors. More precisely, how to show that $G = A \ast B $ is hyperbolic relative to $\{A, B\}$?
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4$\begingroup$ This is almost obvious using the definition of relative hyperbolicity in Osin's paper "Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems", i.e. a linear Dehn function relative to $A$ and $B$. In the case of the free product, this Dehn function is $0$. $\endgroup$– Derek HoltCommented Sep 7, 2023 at 19:31
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$\begingroup$ This is also clear from the definition of relative hyperbolicity from actions on fine hyperbolic graphs: it suffices to consider the action on the Bass-Serre tree. $\endgroup$– AGenevoisCommented Sep 8, 2023 at 5:34
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$\begingroup$ @DerekHolt Hi! Why is the Dehn function is 0 $\endgroup$– IreneCommented Oct 16, 2023 at 18:28
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