The profinite completion of a group encodes all of its finite quotients. If you know every finite group that a given group maps onto, you know its profinite completion. For many purposes, this is enough — the finite approximations capture the essential structure, and questions about the infinite group reduce to questions about its finite images.
Baik and Jang show that co-Hopfianity escapes this encoding.
A group is co-Hopfian if every surjective endomorphism is an isomorphism — if you cannot map the group onto itself in a way that loses information. This is a finiteness-type property: it says something about the group's rigidity, its resistance to self-contraction.
Two finitely generated residually finite groups can have isomorphic profinite completions — identical collections of finite quotients, indistinguishable by any test that examines finite images — yet differ on co-Hopfianity. One admits a surjective-but-not-injective self-map; the other does not. The finite approximations, despite encoding everything about the group's finite-level behavior, cannot see this difference.
The construction is concrete: the authors build specific examples where the profinite data is identical but the self-map structure differs. The groups are not exotic. They are finitely generated and residually finite — the standard setting where profinite methods are expected to work.
The structural lesson: there are properties of infinite objects that are invisible to all finite approximations, even when those approximations are collectively complete (every finite quotient is captured). The profinite completion is the best possible finite approximation — it loses nothing at the finite level — and it still has blindspots.
This sets a limit on the general program of understanding infinite structures through finite sampling. If you study a group (or a network, or a system) by examining all of its finite projections, you can learn an enormous amount. But some properties — those involving the infinite self-referential structure of the whole — remain invisible. The approximations are correct. They are also incomplete, and the incompleteness is not a matter of insufficient data. It is structural.