Goodness of amenable groups and the Disc Embedding Conjecture for 4-manifolds beyond elementary amenable fundamental groups

Freedman's topological 4-manifold machinery — the disc embedding theorem, the surgery exact sequence, and the s-cobordism theorem — is known to be valid for fundamental groups in the class of so-called good groups, which contains all elementary amenable groups and all groups of subexponential growth. Davis and Hillman (2025/2026), building on this, classified compact aspherical 4-manifolds with elementary amenable fundamental group and proved topological rigidity (the Borel Uniqueness Conjecture) for them, while stating that goodness is not known for any amenable group of use here beyond the elementary amenable ones, and that the Borel Uniqueness Conjecture is unknown for every closed aspherical 4-manifold with nonelementary amenable fundamental group. The problem asks whether amenability suffices for goodness: does the Disc Embedding Conjecture hold for every 4-manifold with amenable fundamental group — and, in the source's most optimistic form, for every group with no noncyclic free subgroup — thereby extending surgery, s-cobordism, and Borel uniqueness for aspherical 4-manifolds beyond the elementary amenable class? Both directions are open: no proof beyond the known good classes, and no counterexample (no amenable group is known to be bad).

Background

An aspherical manifold is a manifold whose universal cover is contractible, so the manifold is determined up to homotopy equivalence by its fundamental group $\pi$. The Borel Conjecture asserts that closed aspherical manifolds are topologically rigid: any homotopy equivalence between two such manifolds is homotopic to a homeomorphism (for compact manifolds with boundary, the relative version requires the homotopy to be fixed on the boundary). In dimensions at least 5 this follows from surgery theory for groups satisfying the Farrell–Jones Conjecture (FJC) in L-theory, which identifies the surgery obstruction groups with algebraically computable ones. Dimension 4 is different: surgery must be replaced by Freedman's topological machinery, and its validity depends on the fundamental group.

The engine of that machinery is the disc embedding theorem. In dimensions at least 5, the Whitney trick converts pairs of double points of opposite sign in an immersed disc into an embedded disc by tubing along a Whitney disc; in a 4-manifold the Whitney disc itself can have double points, and repairing them requires the fundamental group to satisfy a condition. A group $\pi$ is called good (Freedman–Quinn; formalized as Definition 12.12 in the volume The Disc Embedding Theorem) precisely when the machinery — the disc embedding theorem, the topological surgery sequence, and the topological s-cobordism theorem — is valid for 4-manifolds with fundamental group $\pi$. The Disc Embedding Conjecture (DEC) is the assertion that the disc embedding theorem holds with no restriction on the fundamental group; proving it for a given class of groups is the same as proving goodness for that class. Modern accounts of the disc embedding theorem for good groups are given in the book by Behrens–Kalmár–Kim–Powell–Ray and by Powell–Ray–Teichner.

The known good groups are: all elementary amenable groups — the smallest class of groups containing the finite and the abelian groups and closed under taking subgroups, quotients, group extensions, and directed unions; in particular all solvable groups — proved by Freedman and Quinn; and all groups of subexponential growth, proved by Freedman–Teichner and by Krushkal–Quinn. Goodness passes to subgroups, quotients, extensions, and filtered colimits. Beyond this, almost nothing is known: as Arunima Ray's 2024 survey puts it, it is not even known whether nonabelian free groups are good, and no group is known to fail goodness.

An amenable group is one admitting a finitely additive, left-invariant probability measure on all subsets, equivalently a Følner sequence of finite sets that are almost invariant under left translation. All elementary amenable groups and all groups of subexponential growth are amenable, but the amenable class is much wider, and the known-good classes together with the closure operations do not reach all amenable groups — amenable groups of exponential growth outside the elementary amenable class are covered by no known goodness result.

In this landscape, Davis and Hillman classified compact aspherical 4-manifolds with elementary amenable fundamental group: the group must be polycyclic (admitting a finite chain of subgroups whose successive quotients are cyclic) or a solvable Baumslag–Solitar group $BS(1,m)$ (presented as $\langle a,b \mid bab^{-1}=a^{m}\rangle$), the possible boundary 3-manifolds are classified, and since these groups are good and satisfy the FJC, such manifolds are topologically rigid in the relative sense of the Borel Uniqueness Conjecture. They emphasize that among fundamental groups of compact aspherical 4-manifolds, the currently known good ones are exactly the elementary amenable ones — no finitely presented group of intermediate growth with finite cohomological dimension is known to exist, so subexponential growth contributes nothing new inside this class of groups. Related rigidity results are all conditional on goodness: for instance Kasprowski and Land prove homeomorphism classification of closed topological 4-manifolds with 4-dimensional fundamental group assuming the group is good and satisfies the FJC. Davis–Hillman state that it is plausible that the DEC holds for all amenable groups, with the most optimistic expectation being that it holds for all groups with no noncyclic free subgroup; they also note that the Borel Uniqueness Conjecture is unknown for any closed aspherical 4-manifold with nonelementary amenable fundamental group. A counterexample to the smooth analogue of Borel uniqueness in dimension 4 (closed aspherical smooth 4-manifolds that are homeomorphic but not diffeomorphic, by Davis–Hayden–Huang–Ruberman–Sunukjian) sharpens the contrast: the topological statement remains fully open outside the elementary amenable class.

Problem Statement

Is every amenable group good — that is, does the Disc Embedding Conjecture hold for every topological 4-manifold with amenable fundamental group, so that the 4-dimensional disc embedding theorem, the topological surgery sequence, and the s-cobordism theorem are valid for all amenable fundamental groups? Davis and Hillman, whose classification of compact aspherical 4-manifolds with elementary amenable fundamental group and proof of topological rigidity for them currently stop exactly at the boundary of the known good groups, state that this extension is plausible for all amenable groups and that the most optimistic expectation is goodness for all groups with no noncyclic free subgroup; they also state that the Borel Uniqueness Conjecture (relative topological rigidity of compact aspherical 4-manifolds) is unknown for every closed aspherical 4-manifold with nonelementary amenable fundamental group.

References

1. James F. Davis and J. A. Hillman, "Aspherical 4-manifolds with elementary amenable fundamental group", 2025 (arXiv v2 2026), arXiv:2501.12512, DOI 10.48550/arxiv.2501.12512, https://arxiv.org/abs/2501.12512 (source paper)
2. Arunima Ray, "Constructing locally flat surfaces in 4-manifolds", 2024, arXiv:2412.18423, DOI 10.48550/arxiv.2412.18423, https://arxiv.org/abs/2412.18423 (survey)
3. Michael H. Freedman and Frank Quinn, "Topology of 4-Manifolds (PMS-39)", Princeton University Press, Princeton Mathematical Series 39, 1990, DOI 10.1515/9781400861064, https://doi.org/10.1515/9781400861064
4. Michael H. Freedman and Peter Teichner, "4-Manifold topology I: Subexponential groups", Inventiones mathematicae 122 (1995) 509–529, DOI 10.1007/BF01231454, https://doi.org/10.1007/BF01231454
5. Vyacheslav S. Krushkal and Frank Quinn, "Subexponential groups in 4-manifold topology", Geometry & Topology 4 (2000) 407–430, DOI 10.2140/gt.2000.4.407, https://doi.org/10.2140/gt.2000.4.407
6. Min Hoon Kim, Patrick Orson, JungHwan Park, and Arunima Ray, "Good Groups", chapter in The Disc Embedding Theorem, Oxford University Press, 2021, DOI 10.1093/oso/9780198841319.003.0019, https://doi.org/10.1093/oso/9780198841319.003.0019
7. Stefan Behrens, Boldizsár Kalmár, Min Hoon Kim, Mark Powell, and Arunima Ray (eds.), "The Disc Embedding Theorem", Oxford University Press, 2021, DOI 10.1093/oso/9780198841319.001.0001, https://doi.org/10.1093/oso/9780198841319.001.0001
8. Mark Powell, Arunima Ray, and Peter Teichner, "The 4-dimensional disc embedding theorem and dual spheres", 2020 (arXiv v3 2025), arXiv:2006.05209, DOI 10.48550/arxiv.2006.05209, https://arxiv.org/abs/2006.05209
9. Daniel Kasprowski and Markus Land, "Topological 4-manifolds with 4-dimensional fundamental group", 2020, arXiv:2007.03399, DOI 10.48550/arxiv.2007.03399, https://arxiv.org/abs/2007.03399
10. James F. Davis and J. A. Hillman, "Aspherical manifolds with boundary", 2025, arXiv:2501.12509, https://arxiv.org/abs/2501.12509
11. James F. Davis and J. A. Hillman, "Aspherical PD_4-pairs", 2026, arXiv:2608.21717, https://arxiv.org/abs/2608.21717
12. Michael Davis, Kyle Hayden, Jingyin Huang, Daniel Ruberman, and Nathan Sunukjian, "Exotic aspherical 4-manifolds", 2024 (arXiv v3 2026), arXiv:2411.19400, DOI 10.48550/arxiv.2411.19400, https://arxiv.org/abs/2411.19400