The generalized conjugacy problem for virtually free groups

Abstract The generalized conjugacy problem (has g a conjugate in K for K rational?) is solved for finitely generated (f.g.) virtually free groups with constraints that go beyond the context-free level, a new result for the free group itself. Moldavanskii's theorem on simultaneous conjugacy of f.g. subgroups of a free group is also generalized for virtually free groups and this wider class of constraints. The solution set of the equation x –1 g φ ( x ) ∈ K in the free group ( φ a virtually inner automorphism, K rational) is shown to be rational and effectively constructible, and a similar result is proved for the equation xgx –1 ∈ K in a f.g. virtually free group. The twisted conjugacy problem with context-free constraints is also proved to be decidable for the free group.