Abstract
A finitely generated residually finite group G is an OEˆ-group if any action of its profinite completion Gˆ on a profinite tree with finite edge stabilizers admits a global fixed point. In this paper, we study the profinite genus of free products G1⁎HG2 of OEˆ-groups G1,G2 with finite amalgamation H. Given such G1,G2,H we give precise formulas for the number of isomorphism classes of G1⁎HG2 and of its profinite completion. We compute the genus of G1⁎HG2 and list various situations when the formula for the genus simplifies.
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