Profinite genus of free products with finite amalgamation

A finitely generated residually finite group $G$ is an $\widehat{OE}$-group if any action of its profinite completion $\widehat{G}$ on a profinite tree with finite edge stabilizers admits a global fixed point. In this paper, we study the profinite genus of free products $G_1{\ast }_H{G}_2$ of $\widehat{OE}$-groups $G_1,G_2$ with finite amalgamation $H$. Given such $G_1,G_2,H$ we give precise formulas for the number of isomorphism classes of $G_1{\ast }_H{G}_2$ and of its profinite completion. We compute the genus of $G_1{\ast }_H{G}_2$ and list various situations when the formula for the genus simplifies.