Cubical residual complexes and profinite rigidity of real hyperbolic lattices

Abstract

We study profinite rigidity of real hyperbolic lattices, with the main application in dimension three. The basic tool is a profinite version of the dual cube complex of a pair of filling curves on a surface, which we call a profinite cubical residue complex. We prove that an equivariant isomorphism between the completions of two such complexes is induced, after conjugation by a single element, by an isomorphism of the discrete groups and complexes, and that the same holds for free cocompact cellular actions of finitely generated residually finite groups on connected, locally finite, finite dimensional regular CW complexes. On a closed orientable surface of genus at least two, a theory of divisible fillings determines the completed cut tree of a nonseparating simple closed curve. Combined with the homological intersection criterion of Boggi and Zalesskii, this determines the completed dual square complex of a nonseparating filling pair, and hence realizes every isomorphism of profinite surface groups which preserves the two marked cyclic subgroups. In dimension three we combine these results with virtual fibering, Liu's profinite invariants of hyperbolic $3$-manifolds and the goodness of $3$-manifold groups. Massey products and virtual domination give cohomological integrality, and pseudo-Anosov dynamics, periodic orbit traces and cross sections of suspension flows give a correspondence of prime periodic orbits together with the fiber curves needed for the realization. Profinite rigidity of closed and of cusped lattices follows, the cusped case through cyclic orbifold fillings, and we obtain $\Out(\Gamma)\cong\Out(\wh\Gamma)$ for every lattice of $\PSL_2(\C)$.