Goodness of hyperbolic and CAT(0)-groups and the Borel conjecture in dimension four

Bartels and Lück (Annals of Mathematics, 2012) proved the Borel conjecture in dimensions at least five for the class $\mathcal{B}$ of groups generated by word-hyperbolic groups and groups acting properly, cocompactly and isometrically on finite-dimensional CAT(0) spaces, and closed under subgroups, finite products, free products, an extension condition, and directed colimits. The two concrete technical gaps left by that paper — the full K-theoretic Farrell–Jones conjecture for CAT(0)-groups and the deferred transfer reducibility of finite-dimensional CAT(0)-groups (a controlled-topology condition on the group's action implying the Farrell–Jones conjecture for it) — were closed by later work (Wegner 2012; Bartels–Lück 2012), so the K- and L-theoretic assembly input now holds in all degrees for every group in $\mathcal{B}$. The single surviving obstruction to extending topological rigidity of aspherical manifolds with fundamental group in $\mathcal{B}$ down to dimension four is Freedman's goodness: whether the 4-dimensional disk embedding theorem, topological surgery, and the s-cobordism theorem hold for compact topological 4-manifolds whose fundamental group lies in $\mathcal{B}$. Since $\mathcal{B}$ contains the finitely generated free groups, which are universal for 4-dimensional surgery, this problem is equivalent in strength to the general 4-dimensional topological surgery conjecture.

Background

A closed manifold $M$ is called aspherical if its universal cover is contractible; equivalently, $M$ is a classifying space $K(G,1)$ for its fundamental group $G=\pi_1(M)$. The Borel conjecture asserts that closed aspherical manifolds are topologically rigid: if $M$ and $N$ are closed aspherical manifolds with isomorphic fundamental groups, then $M$ and $N$ are homeomorphic, and every homotopy equivalence $M\to N$ is homotopic to a homeomorphism. The conjecture is known in dimensions at most three (by surface classification and by Thurston geometrization, established by Perelman), and in dimensions at least five it is approached through surgery theory: the Sullivan–Wall surgery exact sequence reduces rigidity for $G$ to vanishing of the Whitehead group $\mathrm{Wh}(G)$ and of the lower algebraic K-groups of $\mathbb{Z} G$, together with isomorphism of the algebraic L-theory assembly map. Both K- and L-theoretic statements are predicted by the Farrell–Jones conjecture (FJC), which asserts that the assembly maps for the algebraic K- and L-theory of the group ring $\mathbb{Z}G$, indexed by the classifying space for the family of virtually cyclic subgroups (subgroups that are finite or virtually infinite cyclic), are isomorphisms.

Bartels and Lück proved the L-theoretic FJC and the low-degree K-theoretic FJC for the class $\mathcal{B}$: the smallest class of groups that contains all word-hyperbolic groups (finitely generated groups whose Cayley graph is $\delta$-hyperbolic, meaning geodesic triangles are uniformly thin, the standard notion of negative curvature in geometric group theory) and all finite-dimensional CAT(0)-groups (groups acting properly, cocompactly and isometrically by isometries on a finite-dimensional CAT(0) space, a geodesic metric space in which every triangle is at least as thin as its comparison triangle in the Euclidean plane — the standard model of non-positive curvature), and that is closed under passage to subgroups, finite direct products, finite free products, directed colimits with arbitrary structure maps, and the following extension-type condition: if $\pi\colon G\to H$ is a homomorphism with $H\in\mathcal{B}$ and $\pi^{-1}(V)\in\mathcal{B}$ for every virtually cyclic subgroup $V\le H$, then $G\in\mathcal{B}$. This yields the Borel conjecture in dimensions at least five for torsion-free groups in $\mathcal{B}$, covering hyperbolic groups, finite-dimensional CAT(0)-groups such as Coxeter groups and right-angled Artin groups, their subgroups, products and colimits, and $S$-arithmetic groups (groups obtained from arithmetic groups by inverting finitely many primes, such as $\mathrm{SL}_n(\mathbb{Z}[1/p])$).

Dimension four is exceptional. In the topological category, 4-dimensional surgery and the s-cobordism theorem rest on Freedman's disk embedding theorem, whose proof converts an immersed family of disks with dual spheres into embedded disks; in dimension four the Whitney trick of high-dimensional topology fails, because Whitney disks cannot be made disjoint by general position, and the proof instead passes to infinite iterative constructions (capped gropes and Casson handles) whose convergence depends on the fundamental group. A group $G$ is called good (in the sense of Freedman) if the disk embedding theorem — and consequently 4-dimensional topological surgery, the topological s-cobordism theorem, and topological rigidity statements such as the Borel conjecture in dimension four — holds for compact topological 4-manifolds with fundamental group $G$. Known good groups include the elementary amenable groups and all groups of subexponential growth (Freedman–Teichner 1995; Krushkal–Quinn 2000); the class of good groups is closed under extensions and under directed colimits, and it contains $\mathbb{Z}$ and all finite groups. Nothing is known for non-amenable groups of exponential growth: it is open whether the finitely generated free groups, and hence the hyperbolic groups, the right-angled Artin groups, and the finite-dimensional CAT(0)-groups inside $\mathcal{B}$, are good.

A decisive structural fact is that the general 4-dimensional topological surgery conjecture reduces to universal surgery problems whose model fundamental groups are free (Casson–Freedman atomic surgery problems; see Freedman–Krushkal 2020). Since finitely generated free groups are word-hyperbolic and therefore lie in $\mathcal{B}$, establishing goodness for all groups in $\mathcal{B}$ is equivalent in strength to establishing 4-dimensional topological surgery for all groups. Meanwhile the algebraic input is complete: by later work of Wegner (full K-theoretic FJC for CAT(0)-groups) combined with the hyperbolic case of Bartels–Lück–Reich and the inheritance properties of the Farrell–Jones class, every group in $\mathcal{B}$ now satisfies the Full Farrell–Jones Conjecture in K- and L-theory in all degrees. Goodness is therefore the only remaining obstruction to the Borel conjecture in dimension four for the class $\mathcal{B}$.

Problem Statement

Let $\mathcal{B}$ be the smallest class of groups containing all word-hyperbolic groups and all groups acting properly, cocompactly and isometrically on a finite-dimensional CAT(0) space, and closed under: (a) passage to subgroups; (b) finite direct products; (c) finite free products; (d) directed colimits with arbitrary structure maps; and (e) the condition that if $\pi\colon G\to H$ is a homomorphism with $H\in\mathcal{B}$ and $\pi^{-1}(V)\in\mathcal{B}$ for every virtually cyclic subgroup $V\le H$, then $G\in\mathcal{B}$.

Determine whether every group in $\mathcal{B}$ is good in the sense of Freedman; that is, whether the 4-dimensional disk embedding theorem — and consequently 4-dimensional topological surgery, the topological s-cobordism theorem, and the Borel conjecture in dimension four — holds for compact topological 4-manifolds whose fundamental group lies in $\mathcal{B}$. A positive answer would in particular imply — since the Full Farrell–Jones Conjecture in K- and L-theory is now known for every group in $\mathcal{B}$ — that any two closed aspherical topological 4-manifolds with fundamental group isomorphic to a common torsion-free group $G\in\mathcal{B}$ are homeomorphic, with every homotopy equivalence between them homotopic to a homeomorphism.

A positive answer for the whole class $\mathcal{B}$ is equivalent, via the Casson–Freedman universal surgery models, to the general 4-dimensional topological surgery conjecture, because $\mathcal{B}$ contains the finitely generated free groups.

References

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