Genus for groups

We introduce a notion of a genus g(C,G) for a class of groups C and G∈C. It consists of isomorphism classes of groups from C having the same profinite completion as G. We show finiteness results for g(C,G) for several important families of groups including finitely generated virtually free groups. We develop formulas for the number of elements in g(C,G) in various cases. By these we find interesting examples where g(C,G) contains only one element.