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We prove that for any two continuous minimal (topologically free) actions ofthe infinite dihedral group on an infinite compact Hausdorff space, they arecontinuously orbit equivalent only if they are conjugate. We also show theabove fails if we replace the infinite dihedral group with certain othervirtually cyclic groups, e.g. the direct product of the integer group with anynon-abelian finite simple group.

On continuous orbit equivalence rigidity for virtually cyclic group actions