Profinite completion captures a group's finite quotients — all the information visible through homomorphisms to finite groups. Two groups with isomorphic profinite completions share every property detectable by finite approximation. The question: which properties survive profinite completion?
Stable commutator length does not (arXiv:2603.12095). Two groups can have isomorphic profinite completions — identical finite quotient structure — while having completely different stable commutator lengths. The property that measures how efficiently elements can be expressed as products of commutators is invisible to all finite approximations of the group.
The proof constructs Grothendieck pairs — pairs of groups where one embeds in the other and the embedding induces an isomorphism on profinite completions, yet the groups differ on the target property. The construction combines Rips constructions (which produce groups with wild properties from well-behaved quotients) with iterated group-theoretic Dehn filling on hyperbolic virtually special groups.
The casualties extend beyond stable commutator length. Quasimorphisms — functions that are “almost” homomorphisms — are not profinite invariants, answering a question of Echtler and Kammeyer. Property NL (obstructing actions on hyperbolic spaces) and property FW_infinity (obstructing actions on finite-dimensional CAT(0) cube complexes) are also invisible. Even the existence of non-abelian free subgroups is not detected.
The pattern: geometric and dynamical properties of groups — how they act on spaces, how their elements decompose — are systematically invisible to finite approximation. Profinite completion sees the algebra. It is blind to the geometry.