Wall's conjecture for Poincaré duality groups of dimension 3

The 2010 monograph 'Geometrisation of 3-Manifolds' by Bessières, Besson, Boileau, Maillot and Porti closes its introductory survey with a list of open problems, including a canonical surgery procedure for Ricci flow with bubbling-off (Remark 1.2.4), linearity of fundamental groups of closed 3-manifolds, and the group-theoretic recognition questions of Section 1.4.2, among them Wall's conjecture for PD(3)-groups. Wall's conjecture predicts that every finitely presented Poincaré duality group of dimension 3 is the fundamental group of a closed aspherical 3-manifold. The conjecture remains completely open — in particular for the atoroidal and hyperbolic cases, where it embeds the Cannon conjecture — despite substantial partial results from geometrisation, JSJ-type splitting theorems, and recent quasimorphism rigidity evidence. The Ricci-flow sub-question from the same source list has since been resolved affirmatively — Kleiner and Lott constructed singular Ricci flows and Bamler and Kleiner proved their uniqueness and stability, yielding a canonical Ricci flow through singularities; the linearity question is narrowed but still open for closed graph manifolds.

Background

A group $G$ is a Poincaré duality group of dimension $n$ (a PD( $n$ )-group) if it is of type FP (the trivial $\mathbb{Z}G$ -module $\mathbb{Z}$ admits a finite-length resolution by finitely generated projective $\mathbb{Z}G$ -modules) and $H^n(G;\mathbb{Z}G)\cong\mathbb{Z}$ while $H^i(G;\mathbb{Z}G)=0$ for $i\neq n$ , where $\mathbb{Z}G$ is the integral group ring and $H^*(G;\mathbb{Z}G)$ is cohomology of $G$ with coefficients in it. This axiomatizes, purely algebraically, the Poincaré duality with local coefficients satisfied by the fundamental group of a closed aspherical $n$ -manifold (a closed manifold whose universal cover is contractible); every such group is a PD( $n$ )-group, and every PD( $n$ )-group is torsion-free and admits an 'orientable' subgroup of index at most 2. Wall asked, in work on Poincaré complexes culminating in his 1967 paper and in Question G2 of the 1979 LMS volume 'Homological Group Theory', whether this duality axiom already characterizes manifold groups among finitely presented groups; in dimension 3 the question has become known as Wall's conjecture: every finitely presented PD(3)-group is the fundamental group of a closed aspherical 3-manifold. For $n=2$ the answer is affirmative — PD(2)-groups are precisely surface groups (Eckmann–Linnell–Müller) — and Davis constructed, for every $n\geq 4$ , PD( $n$ )-groups that are not finitely presentable and hence are not manifold groups, so finite presentability is an essential hypothesis; for finitely presented groups the question is open in every dimension $n\geq 3$ . Dimension 3 is the critical open case, because there the question meets Thurston's geometrisation conjecture, proved by Perelman via Ricci flow with surgery: geometrisation decomposes every closed 3-manifold into Seifert fibered pieces (which fiber over a 2-orbifold with circle fibers) and hyperbolic pieces (whose interiors carry complete finite-volume hyperbolic structures), and a Kneser–Milnor prime decomposition, giving a nearly complete topological classification. A first consequence is the observation of Groves–Manning–Wilton that a PD(3)-group is a 3-manifold group if and only if it is virtually one (that is, contains a finite-index subgroup that is), so Wall's conjecture is exactly the statement that 'virtually geometric implies geometric' for these duality groups. The known partial results all exploit some algebraic trace of geometry: the Dunwoody–Swenson algebraic torus theorem (refined by Castel's JSJ-type splitting for PD(3)-pairs) produces splittings over $\mathbb{Z}^2$ subgroups analogous to the JSJ decomposition; the conjecture holds for PD(3)-groups containing a nontrivial normal cyclic subgroup (Hillman when the first Betti number is positive, Bowditch in general — such groups are Seifert fibered manifold groups) and, by work of Hillman and Thomas, for groups with an almost finitely presented normal subgroup of infinite index. The atoroidal case — no $\mathbb{Z}^2$ subgroups — is untouched, and it contains the hyperbolic face of the problem: a word-hyperbolic group (one whose Cayley graph is $\delta$ -hyperbolic) has a canonical compactification by its Gromov boundary $\partial G$ , and Bestvina–Mess showed that Wall's question for torsion-free hyperbolic PD(3)-groups is equivalent to the Cannon conjecture, which predicts that a hyperbolic group with $\partial G\cong S^2$ is virtually Kleinian (a Kleinian group is a discrete subgroup of $\mathrm{PSL}(2,\mathbb{C})$ ; equivalently, virtually the fundamental group of a closed hyperbolic 3-manifold). The Cannon conjecture is itself open: Markovic reduced it to producing enough quasiconvex surface subgroups, Ferry–Lück–Weinberger proved a stable version, Cheetham-West–Nolte characterized its candidates via geometric measure theory, and Bregman–Incerti-Medici settled a cubulated generalization. Even the basic finiteness prerequisites of Wall's conjecture are open: it is not known whether every PD(3)-group is finitely presented, and no PD(3)-group is known to be non-coherent (coherent meaning every finitely generated subgroup is finitely presented). A linearity question recorded next to these in the source monograph — whether fundamental groups of all closed 3-manifolds admit faithful finite-dimensional complex representations — has been narrowed by geometrisation and virtual specialness to the case of closed graph manifolds (all of whose JSJ pieces are Seifert fibered).

Problem Statement

Wall's conjecture for 3-dimensional Poincaré duality groups. Prove or disprove: every finitely presented PD(3)-group — that is, a group $G$ of type FP with $H^3(G;\mathbb{Z}G)\cong\mathbb{Z}$ and $H^i(G;\mathbb{Z}G)=0$ for $i\neq 3$ , taken orientable up to passing to an index-2 subgroup — is isomorphic to the fundamental group of a closed aspherical 3-manifold (equivalently, when $G$ is orientable, of a closed orientable irreducible 3-manifold). An affirmative answer must cover all finitely presented PD(3)-groups, including the atoroidal and hyperbolic cases where no current technique reaches. A negative answer must exhibit one finitely presented PD(3)-group together with a proof that it is not isomorphic to the fundamental group of any closed 3-manifold. By the geometrisation theorem it would suffice — and it is necessary — to decide whether every finitely presented PD(3)-group is virtually the fundamental group of a closed 3-manifold. Answers restricted to PD(3)-groups with extra structure (a normal cyclic subgroup, a splitting, a cubulation, coherence, etc.) address only special cases, not this problem.

References

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