In a 1970 paper, Grothendieck proved that a homomorphism u:Γ1→Γ2u:\Gamma_1\to\Gamma_2 of finitely generated groups that induces an isomorphism u^:Γ^1→Γ^2\hat{u}:\hat{\Gamma}_1\to\hat{\Gamma}_2 of profinite completions forces an equivalence of the finitely presented representation theories over any ground ring, and asked for conditions on the groups that force uu itself to be an isomorphism. The unrestricted versions of this question were settled negatively: Platonov and Tavgen' (finitely generated groups, 1986/1990), Bridson and Grunewald (finitely presented groups, 2004, whose group Γ\Gamma lies outside Grothendieck's class C0C^0 and whose subgroup PP has infinite index in the Tannaka closure clA(P)\mathrm{cl}_A(P) for every non-zero commutative ring AA), and Lubotzky (1979/1980) with Lubotzky and Venkataramana (2019), who gave an essentially complete criterion for integral linear groups: clZ(Γ)=Γ\mathrm{cl}_{\mathbb{Z}}(\Gamma)=\Gamma if and only if Γ\Gamma has the congruence subgroup property. The surviving open problem, restated by Jaikin-Zapirain and Lubotzky (2025) as left/right Grothendieck rigidity, is to characterize which finitely generated residually finite groups admit no non-trivial Grothendieck pairs and which targets lie in Grothendieck's class CC, and to decide the recognized open instances: left Grothendieck rigidity of 3-manifold groups (right rigidity is known), of non-arithmetic lattices in SL2(C)\mathrm{SL}_2(\mathbb{C}), of lattices in SO(n,1)\mathrm{SO}(n,1) for n≥4n\ge 4, right rigidity of coherent residually finite hyperbolic groups, whether right rigidity implies left rigidity, whether these properties are commensurability invariants, and whether profinite rigidity implies left or right Grothendieck rigidity. This core remains open, with an active 2024-2026 literature producing both new counterexamples and new positive rigidity classes.

Background

In 1970 Grothendieck published a short paper connecting the representation theory of a discrete group with its profinite completion. The profinite completion Γ^\hat{\Gamma} of a group Γ\Gamma is the inverse limit of the directed system of its finite quotients, equipped with a natural map Γ→Γ^\Gamma\to\hat{\Gamma}; the group is called residually finite when this map is injective, i.e. when finite quotients separate the elements of Γ\Gamma. Grothendieck's motivation came from algebraic geometry: for a smooth projective complex variety XX, the profinite completion of the topological fundamental group π1(Xan)\pi_1(X^{an}) - which is finitely presented - coincides with the étale fundamental group of XX, an object defined by purely algebraic means, and he wanted to understand how much of a discrete group is visible to its profinite completion. His main theorem states that if u:Γ1→Γ2u:\Gamma_1\to\Gamma_2 is a homomorphism of finitely generated groups such that the extension u^:Γ^1→Γ^2\hat{u}:\hat{\Gamma}_1\to\hat{\Gamma}_2 is an isomorphism, then for every non-zero commutative ring AA the representation theories agree: the category of finitely presented AA-modules carrying a Γ2\Gamma_2-action is equivalent to the corresponding category for Γ1\Gamma_1. In particular, every representation of Γ1\Gamma_1 into a group of the form G(A)G(A), where GG is an affine group scheme of finite type over a commutative ring, extends uniquely along uu - a superrigidity-type statement.

The paper then raises the problem that dominates its reception, stated in its abstract: find good conditions on the groups which ensure that uu is an isomorphism whenever u^\hat{u} is. A Grothendieck pair is an embedding φ:Λ↪Γ\varphi:\Lambda\hookrightarrow\Gamma of finitely generated residually finite groups (injectivity is automatic from residual finiteness once φ^\hat{\varphi} is an isomorphism) such that φ^:Λ^→Γ^\hat{\varphi}:\hat{\Lambda}\to\hat{\Gamma} is an isomorphism; the pair is trivial when φ\varphi is onto. Grothendieck introduced two devices toward the problem. The first is the class CC of groups KK such that Hom(Γ,K)→Hom(Λ,K)\mathrm{Hom}(\Gamma,K)\to\mathrm{Hom}(\Lambda,K), f↦f∘φf\mapsto f\circ\varphi, is a bijection for every Grothendieck pair Λ↪Γ\Lambda\hookrightarrow\Gamma (with the variant class C0C^0 allowing arbitrary homomorphisms of finitely presented groups that induce isomorphisms of profinite completions). The second is the Tannaka closure clA(Γ)\mathrm{cl}_A(\Gamma) - later developed by Lubotzky as Tannaka duality for discrete groups - defined as the group of tensor-compatible automorphisms of the forgetful functor on the category of finitely generated AA-modules with a Γ\Gamma-action, through which one tries to recover Γ\Gamma from its representation category; a positive answer to the question 'is Γ→clZ(Γ)\Gamma\to\mathrm{cl}_{\mathbb{Z}}(\Gamma) an isomorphism?' would have implied a positive solution of the main problem.

Grothendieck proved that CC is closed under commensurability (passage between subgroups of finite index), inverse limits and direct products, that it contains all nilpotent groups, all compact Hausdorff groups, and all groups G(A)G(A) as above, and that arithmetic groups with the strict congruence subgroup property (the property that every subgroup of finite index contains the kernel of a reduction map, i.e. is a congruence subgroup) satisfy clZ(Γ)≅Γ\mathrm{cl}_{\mathbb{Z}}(\Gamma)\cong\Gamma. Since any Λ∈C\Lambda\in C is left Grothendieck rigid, these results imply, for example, that finitely generated free groups, surface groups and SS-arithmetic groups admit no non-trivial Grothendieck pair on the left.

The naive hopes attached to the problem were destroyed over the following decades: Platonov and Tavgen' produced non-trivial Grothendieck pairs of finitely generated residually finite groups via fibre products (subgroups of direct products built from a short exact sequence), Bridson and Grunewald produced such pairs with both groups finitely presented and simultaneously destroyed the Tannaka-closure strategy, and Lubotzky and Venkataramana determined exactly the integral linear groups for which the strategy survives. In parallel a positive program developed: Long and Reid, then Boileau and Friedl, and finally Sun proved that all finitely generated 3-manifold groups are Grothendieck rigid in the sense that no proper finitely generated subgroup forms a Grothendieck pair with them.

The modern terminology crystallizes the surviving question: a finitely generated residually finite group Λ\Lambda is left Grothendieck rigid (LGR) if every Grothendieck pair Λ↪Γ\Lambda\hookrightarrow\Gamma is trivial, a group Γ\Gamma is right Grothendieck rigid (RGR, called simply 'Grothendieck rigid' in the 3-manifold literature) if every Grothendieck pair Λ↪Γ\Lambda\hookrightarrow\Gamma is trivial, and a group is profinitely rigid if every finitely generated residually finite group with an isomorphic profinite completion is isomorphic to it - a related but distinct property.

Terminology used in the instances of the problem: a 3-manifold group is the fundamental group of a compact 3-dimensional manifold; a lattice in a Lie group is a discrete subgroup of finite covolume; SS-arithmetic groups are matrix groups over rings of integers with finitely many primes inverted; a group is coherent if each of its finitely generated subgroups is finitely presented; a group is word-hyperbolic (Gromov-hyperbolic) if its Cayley graph has uniformly thin geodesic triangles, the standard large-scale negative-curvature condition in geometric group theory; the Bohr compactification of a group is the universal compact Hausdorff group it maps to (equivalently, the completion with respect to all finite-dimensional unitary representations), and its proalgebraic completion is the universal pro-affine algebraic group it maps to; Grothendieck's theorem implies that the two members of any Grothendieck pair have isomorphic Bohr compactifications, which is why Bohr rigidity appears as a weak form of profinite rigidity. A finitely generated residually finite group Λ\Lambda is Bohr rigid (respectively proalgebraic rigid) if every finitely generated residually finite group whose Bohr compactification (respectively proalgebraic completion) is isomorphic to that of Λ\Lambda is isomorphic to Λ\Lambda. The problem recorded here is the characterization problem for these rigidity properties and for the class CC.

Problem Statement

Determine which finitely generated residually finite groups are left Grothendieck rigid and which are right Grothendieck rigid, by intrinsic conditions on the groups themselves - conditions that do not merely restate the hypothesis that the induced map of profinite completions is an isomorphism. 'Intrinsic' here means a property of the group alone (or of its finite-index subgroups and their finite-dimensional representations) that can be stated without reference to a chosen Grothendieck pair or to the specific map of profinite completions inducing it; a criterion merely restates the hypothesis when it is the defining property in disguise, such as 'no proper finitely generated subgroup has the same profinite completion'. The single objective is to characterize, for the natural classes of discrete groups, which groups Λ\Lambda have the property that every Grothendieck pair Λ↪Γ\Lambda\hookrightarrow\Gamma (an embedding φ:Λ↪Γ\varphi:\Lambda\hookrightarrow\Gamma of finitely generated residually finite groups such that φ^:Λ^→Γ^\hat{\varphi}:\hat{\Lambda}\to\hat{\Gamma} is an isomorphism of profinite completions) is trivial and which groups Γ\Gamma have the property that every such Grothendieck pair is trivial, and dually to characterize Grothendieck's class CC of target groups KK for which every Grothendieck pair Λ↪Γ\Lambda\hookrightarrow\Gamma induces a bijection Hom(Γ,K)→Hom(Λ,K)\mathrm{Hom}(\Gamma,K)\to\mathrm{Hom}(\Lambda,K). Concretely, the recognized open instances of this objective are: (a) is every finitely generated 3-manifold group left Grothendieck rigid (right Grothendieck rigidity is known)? (b) is a non-arithmetic lattice in SL2(C)\mathrm{SL}_2(\mathbb{C}) left Grothendieck rigid? (c) are lattices in SO(n,1)\mathrm{SO}(n,1), n≥4n\ge 4, left or right Grothendieck rigid? (d) is every coherent residually finite word-hyperbolic group right Grothendieck rigid? (e) does right Grothendieck rigidity imply left Grothendieck rigidity? (f) are left and right Grothendieck rigidity invariant under passage to commensurable groups (finite-index subgroups and finite extensions)? (g) is every profinitely rigid group left or right Grothendieck rigid? (h) is a finitely generated free group Bohr rigid, and is it proalgebraic rigid? The general characterization, together with the dual characterization of the class CC, is the objective: the eight instances (a)-(h) are its recognized open instances, each independently decidable. This is the direct continuation, in its post-2004 form, of the problem posed in Grothendieck's 1970 paper; the original unrestricted versions - whether all finitely presented residually finite groups are rigid, and whether the Tannaka closure recovers the group universally - are known to have negative answers and are not part of what is asked.

References

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