Which compact 3-manifold groups are RFRS: the strict (non-virtual) clause remaining from Agol's 2008 RFRS question

Agol's 2008 paper introducing the RFRS (residually finite rationally solvable) condition asks: 'Which 3-manifolds have RFRS or virtually RFRS fundamental group?' The virtual clause of this question is now completely resolved — the fundamental group of a compact, orientable, aspherical 3-manifold is virtually RFRS exactly when the manifold is nonpositively curved, by Liu's virtual cubulation of nonpositively curved graph manifolds, the virtual specialness theorems of Agol and Przytycki–Wise, and Leeb's nonpositive-curvature criterion — and the condition is effectively decidable from the JSJ decomposition. The strict clause, however, remains open: RFRS passes to subgroups but not to finite extensions, so 'virtually RFRS' is genuinely weaker (knot complements in $S^3$ are virtually RFRS but not RFRS), and no complete classification of 3-manifold groups that are literally RFRS exists in the literature. The open problem is to give necessary and sufficient conditions, in terms of the geometric decomposition and algebraic invariants of the pieces, under which the fundamental group of a compact 3-manifold is RFRS in the strict (non-virtual) sense.

Background

By the geometrization theorem (Thurston, Perelman), a compact, orientable, irreducible 3-manifold with empty or toroidal boundary splits along essential tori into its JSJ decomposition, whose pieces are either hyperbolic or Seifert fibered; a Seifert fibered space is a 3-manifold foliated by circles, with fundamental group an extension of a surface-orbifold group by $Z$. A manifold all of whose JSJ pieces are Seifert fibered is called a graph manifold. Following the convention of the Aschenbrenner–Friedl–Wilton monograph, such a manifold is nonpositively curved if its interior admits a complete Riemannian metric of nonpositive sectional curvature.

A group $G$ is residually finite rationally solvable (RFRS), a notion introduced by Agol, if there is a chain of finite-index normal subgroups $G=G_0\geq G_1\geq G_2\geq\cdots$ with $\bigcap_i G_i=\{1\}$ such that each $G_{i+1}$ contains the kernel of the natural map $G_i\to H_1(G_i;\mathbb{Q})$. Intuitively, the manifold has a cofinal tower of finite abelian covers that successively kill rational commutators. The condition was discovered as the hypothesis of Agol's virtual fibering criterion: an irreducible 3-manifold with vanishing Euler characteristic whose fundamental group is RFRS is virtually fibered, i.e., has a finite cover that fibers over the circle. Right-angled Artin groups (RAAGs, the groups given by generators that commute according to the edges of a graph) are RFRS, and the property passes to subgroups, so every group that is special in the sense of Haglund–Wise (the fundamental group of a special cube complex, equivalently a subgroup of a RAAG) is RFRS. In the finitely generated case, Okun and Schreve characterize RFRS groups as exactly the groups residually in the class of virtually abelian, locally indicable groups (locally indicable meaning every nontrivial finitely generated subgroup surjects onto $Z$).

In Section 9 of the paper introducing the condition, Agol posed verbatim: 'Question: Which 3-manifolds have RFRS or virtually RFRS fundamental group?' He recorded the known obstructions: Seifert fibered spaces with nonzero Euler class of the fibration over a base of nonzero Euler characteristic admit no finite cover with RFRS fundamental group; Boileau and Wang built examples derived from these that cannot have RFRS fundamental group; Sol and Nil manifolds — Sol being the Thurston geometry of torus bundles over the circle whose gluing monodromy is Anosov (hyperbolic), and Nil that of nontrivial circle bundles over the torus — are virtually fibered but not virtually RFRS; and knot complements cannot have RFRS fundamental group, though they may still be virtually RFRS.

The virtual clause of the question is now completely settled. Building on Daniel Wise's cubulation program, Agol proved that closed hyperbolic 3-manifolds are virtually compact special; Przytycki and Wise proved the same for mixed 3-manifolds (those with both hyperbolic and Seifert fibered JSJ pieces) and for graph manifolds with boundary; and Liu proved that for an aspherical compact graph manifold, nonpositive curvature, virtual homotopy equivalence to a special cube complex, virtual embedding of the fundamental group into a finitely generated RAAG, and virtual RFRS are all equivalent. Since Leeb showed that an irreducible 3-manifold with empty or toroidal boundary that is not a closed graph manifold is nonpositively curved, and the Aschenbrenner–Friedl–Wilton synthesis states that a compact, orientable, aspherical 3-manifold is nonpositively curved if and only if its fundamental group is virtually special, the consequence is a complete effective characterization: $\pi_1(M)$ is virtually RFRS if and only if $M$ is nonpositively curved, a condition decidable from the JSJ data by the work of Leeb and of Buyalo and Svetlov. In particular, knot complements — which Agol had left as 'still possible' — are virtually RFRS, and virtual fibering does not imply virtual RFRS: a nontrivial torus bundle with Sol monodromy fibers but its group is not virtually RFRS.

What survives is the strict, non-virtual clause. RFRS passes to subgroups but not to finite extensions, so 'virtually RFRS' is genuinely weaker than 'RFRS': knot complements in $S^3$ are virtually RFRS but not RFRS, and in the adjacent braid-group setting Wu and Ye recently proved that pure braid groups are RFRS, which makes braid groups virtually RFRS, while whether the braid groups $B_n$ are themselves RFRS remains open for $n\geq 4$ — the trefoil knot group is the braid group $B_3$, and since knot complements in $S^3$ are not RFRS, the $B_3$ case is already settled on the negative side, so the strict/virtual boundary runs straight through 3-manifold groups. The strict clause asks which of the nonpositively curved (virtually RFRS) 3-manifold groups are actually RFRS.

Problem Statement

Give a complete characterization of the compact 3-manifolds $M$ whose fundamental group is RFRS in the strict (non-virtual) sense, in terms of the prime decomposition, the JSJ decomposition, the geometry of the JSJ pieces (hyperbolic or Seifert fibered) or of the whole manifold when it carries Sol geometry, and algebraic invariants of the pieces such as the rational homology behavior of their finite covers. Non-orientable manifolds belong to this class and must be treated in their own right: because RFRS passes to subgroups but not to finite-index supergroups, passing to the orientation double cover does not settle the non-orientable cases.

Free products of RFRS groups are RFRS (Agol's Corollary 2.3), and a nontrivial finite group is never RFRS: its rational abelianization is trivial, so the kernel of the map $G_i\to H_1(G_i;\mathbb{Q})$ is all of $G_i$, and the defining chain cannot descend below the group itself. The reducible case follows from the first fact, the spherical prime case from the second; the aspherical prime case, across all geometries, closed and with boundary, follows from neither. In the aspherical prime case, $\pi_1(M)$ RFRS implies $M$ nonpositively curved, which by the resolved virtual clause is equivalent to $\pi_1(M)$ being virtually special and virtually RFRS; knot complements in $S^3$ are not RFRS although they are virtually RFRS; Sol and Nil manifolds, and closed Seifert fibered spaces with nonzero rational Euler class of the fibration over a base orbifold of nonzero Euler characteristic, are not even virtually RFRS (such a closed Seifert fibered space is not virtually fibered, while virtual RFRS with vanishing Euler characteristic would force virtual fibering); and every 3-manifold group that embeds in a right-angled Artin group is RFRS.

References

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