Virtual splitting of the center of the fundamental group for complete finite-volume manifolds of nonpositive curvature without finite generation

The center theorem of Lawson and Yau (1972) describes the center $\mathcal{Z}$ of the fundamental group of a compact manifold of nonpositive sectional curvature: $\mathcal{Z}$ is free abelian of finite rank $k$, equal to the number of linearly independent parallel vector fields, and an abelian Riemannian covering $T^k \times M' \to M$ exhibits the central part of $\pi_1(M)$ geometrically. Their finite-volume extension, and the companion result (Corollary 6) that a finite-index subgroup $\pi'$ realizes the classifying fibration $K(\mathcal{Z},1) \to K(\pi',1) \to K(\pi'/\mathcal{Z},1)$ as a differentiable fiber bundle, rest on the hypothesis that $\pi_1(M)$ be finitely generated, which enters the proofs through a commutator-bound lemma and constancy arguments for convex displacement functions. Complete finite-volume nonpositively curved manifolds with infinitely generated fundamental group exist in every dimension at least 2, and the later literature has never removed the hypothesis: Schroeder's finite-volume splitting theorem covers product group actions, Ballmann and Eberlein analyzed the fundamental group only under a lower curvature bound, and the CAT(0) lattice theory of Caprace and Monod proves the analogous virtual splitting only for finitely generated lattices, its authors explicitly stating they do not know their conclusions without finite generation. The problem is to determine which of the Lawson-Yau finite-volume conclusions survive without any finite-generation assumption - in particular whether the center is still $\mathbb{Z}^k$ and whether the virtual center-splitting with differentiable-fiber-bundle realization holds - or to construct a counterexample.

Background

A complete connected Riemannian manifold $M$ has nonpositive sectional curvature if every tangent two-plane has curvature $K \le 0$. By the Cartan-Hadamard theorem the universal cover $\widetilde{M}$ is then a Hadamard manifold (simply connected, nonpositively curved, hence diffeomorphic to $\mathbb{R}^{\dim M}$), and $M = \widetilde{M}/\pi_1(M)$ is the quotient by the deck group $\pi_1(M)$, which acts freely and properly discontinuously by isometries. $M$ has finite volume if $\mathrm{vol}(M) < \infty$; when $M$ is noncompact this allows cuspidal ends with arbitrarily fast volume decay. A group is finitely generated if some finite subset generates it. Bishop and O'Neill (1969) constructed the first complete noncompact finite-volume quotients of negatively curved spaces, and subsequent work (Nguyen Phan 2011; Avramidi and Nguyen Phan 2021) produced complete finite-volume manifolds of nonpositive curvature whose fundamental group is not finitely generated, in every dimension at least 2, with bounded sectional curvature in the latter construction.

Two classical theorems organize the compact case. The flat torus theorem (Gromoll and Wolf 1971; Lawson and Yau 1972) states that if $\pi_1(M)$ contains a free abelian subgroup of rank $k$, then $\widetilde{M}$ contains a $\pi_1(M)$-invariant convex flat subspace isometric to $\mathbb{R}^k$ on which that subgroup acts cocompactly by translations, so $M$ contains an immersed totally geodesic flat $k$-torus. The center theorem of Lawson and Yau describes the center $\mathcal{Z}$ of $\pi_1(M)$, the subgroup of elements commuting with every element: for compact $M$, $\mathcal{Z}$ is a finitely generated free abelian group $\mathbb{Z}^k$, the number of linearly independent parallel vector fields on $M$ (vector fields covariantly constant under the Levi-Civita connection) equals $k$, and $M$ admits an abelian Riemannian covering $T^k \times M' \to M$ by a product of a flat $k$-torus with a nonpositively curved manifold $M'$. Related exact sequences express $\pi_1(M)$ as an extension of the abelian deck group $\mathcal{A}$ by $\mathcal{Z} \times \mathcal{N}$ with $\mathcal{N} = \pi_1(M')$, and control first homology.

Since every complete nonpositively curved manifold is aspherical (its universal cover is contractible), it is an Eilenberg-MacLane space $K(\pi,1)$: the unique-up-to-homotopy aspherical space with fundamental group $\pi$, the natural classifying space for $\pi$. A short exact sequence of groups $1 \to \mathcal{Z} \to \pi' \to \pi'/\mathcal{Z} \to 1$ induces a fibration $K(\mathcal{Z},1) \to K(\pi',1) \to K(\pi'/\mathcal{Z},1)$ of classifying spaces. When $\mathcal{Z} \cong \mathbb{Z}^k$, $K(\mathcal{Z},1)$ is the $k$-torus, and asking whether the homotopy-theoretic fibration is realized as a differentiable fiber bundle asks whether it comes from a smooth bundle of manifolds with torus fibers.

Lawson and Yau extended their results to complete manifolds of finite volume; finite generation of $\pi_1(M)$ is used in the proofs of the finite-volume results and is stated as a hypothesis of the companion theorems and of Corollary 6, which produces a finite-index subgroup $\pi'$ containing $\mathcal{Z}$ for which the classifying fibration is realized by a differentiable fiber bundle. The hypothesis enters the proofs concretely: a lemma bounding the number of distinct commutators, used to show that deck transformations acting trivially on the Euclidean factor of a de Rham splitting of $\widetilde{M}$ have a power in the center, and constancy arguments for convex displacement functions $p \mapsto d^2(p, g p)$ on the quotient, which are automatic for compact quotients (Bishop and O'Neill) but not evidently so for noncompact finite-volume ones. The question is whether the hypothesis is genuinely needed.

Subsequent structure theory sharpened the compact picture without covering this gap. Eberlein (1983) showed for compact quotients that the Euclidean factor of the de Rham splitting $\widetilde{M} \cong \mathbb{R}^k \times \widetilde{M}'$ is detected by $\pi_1(M)$. Rank rigidity (Ballmann, Brin and Eberlein 1985) implies for compact $M$ that the irreducible de Rham factors of $\widetilde{M}$ are symmetric spaces of noncompact type or rank-one spaces, with later work extending the picture to finite volume. Schroeder (1985) generalized the Gromoll-Wolf/Lawson-Yau product splitting theorem to finite volume: a deck group splitting as a product forces an invariant convex product splitting of the universal cover. Ballmann and Eberlein (1987) proved for finite-volume manifolds with nonpositive curvature bounded below that either $M$ is flat or $\pi_1(M)$ contains a nonabelian free subgroup, and that the geometric rank of $M$ (the minimal dimension of a space of parallel Jacobi fields along geodesics) equals the algebraic rank of $\pi_1(M)$ (the maximal rank of a free abelian subgroup). None of these addresses the center or the virtual splitting for groups that are not finitely generated.

The modern abstraction is the theory of lattices in proper CAT(0) spaces: a CAT(0) space is a geodesic metric space whose triangles are at most as fat as Euclidean comparison triangles (Hadamard manifolds are exactly the Riemannian examples), and a lattice is a discrete group of isometries with finite-volume quotient. Caprace and Monod (2009) proved the CAT(0) analog of the virtual splitting: if a finitely generated lattice acts on a proper CAT(0) space whose isometry group acts cocompactly, then the lattice has a finite-index subgroup splitting as $\mathbb{Z}^n \times \Gamma'$ with the $\mathbb{Z}^n$ acting as a lattice of translations on the Euclidean factor. They retain the finite-generation hypothesis throughout, remark explicitly that they do not know their conclusions without it, and their framework requires an ambient cocompact group - a condition that fails for a general finite-volume manifold, whose universal cover's full isometry group need not act cocompactly. The question thus sits precisely in the gap left open by both the classical Riemannian theory and the modern CAT(0) theory.

Problem Statement

Let $M$ be a complete connected Riemannian manifold of finite volume and nonpositive sectional curvature, let $\pi = \pi_1(M)$, and let $\mathcal{Z}$ be the center of $\pi$. The source question of Lawson and Yau (1972) is what survives of their finite-volume conclusions when the hypothesis that $\pi$ be finitely generated is dropped: finite generation is used in the proofs of the finite-volume results and is stated as a hypothesis of the companion theorems and of the fibration corollary (Corollary 6). It is precisely in the setting where $\pi$ is not finitely generated (examples exist in every dimension at least 2) that the question has content; a trivial center makes the fibration conclusion vacuous, so the content of the question lies in manifolds with nontrivial center. Determine, without any finite-generation assumption:

(1) whether the finite-volume center theorem conclusions hold: $\mathcal{Z}$ is a finitely generated free abelian group $\mathbb{Z}^k$; $M$ is foliated by totally geodesic flat $k$-tori; there is an abelian Riemannian covering $T^k \times M' \to M$ by a product with $\mathcal{N} = \pi_1(M')$, fitting into exact sequences $1 \to \mathcal{Z} \times \mathcal{N} \to \pi_1(M) \to \mathcal{A} \to 1$ and $0 \to \mathcal{Z} \times (\mathcal{N}/[\pi_1(M), \pi_1(M)]) \to H_1(M,\mathbb{Z}) \to \mathcal{A} \to 0$ with $\mathcal{A}$ the abelian deck group;

(2) whether the companion conclusions of the finitely generated theory survive: the equivalence between $k$ and the number of linearly independent globally parallel vector fields on $M$, and the locally free isometric $T^k$-action on $M$ capturing all isometries homotopic to the identity;

(3) the fibration question proper: whether there exists a finite-index subgroup $\pi'$ of $\pi$, containing $\mathcal{Z}$, such that the fibration $K(\mathcal{Z},1) \to K(\pi',1) \to K(\pi'/\mathcal{Z},1)$ can be realized as a differentiable fiber bundle.

The conclusions of (1), (2) and (3) are asserted jointly for the entire class and without any finite-generation hypothesis: a single complete finite-volume nonpositively curved manifold with infinitely generated fundamental group at which any one of them fails - whether the center assertion, the flat-torus foliation, the abelian covering, either exact sequence of (1), the parallel-field count or the torus action of (2), or the fibration realization of (3) - refutes the affirmative answer. A result proved only under an additional hypothesis on the class, such as bounded curvature, a tameness condition on the ends, or a visibility condition on the universal cover (any two distinct points at infinity joined by a geodesic line), is confined to that subclass and is strictly weaker than the question above.

References

1. H. Blaine Lawson, Jr. and Shing Tung Yau, Compact manifolds of nonpositive curvature, Journal of Differential Geometry 7 (1972), 211-228. DOI: 10.4310/jdg/1214430828. URL: https://doi.org/10.4310/jdg/1214430828
2. Detlef Gromoll and Joseph A. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bulletin of the American Mathematical Society 77 (1971), 545-552. DOI: 10.1090/S0002-9904-1971-12747-7. URL: https://doi.org/10.1090/S0002-9904-1971-12747-7
3. R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Transactions of the American Mathematical Society 145 (1969), 1-49. DOI: 10.1090/s0002-9947-1969-0251664-4. URL: https://doi.org/10.1090/s0002-9947-1969-0251664-4
4. Patrick Eberlein, Euclidean de Rham factor of a lattice of nonpositive curvature, Journal of Differential Geometry 18 (1983), 209-220. DOI: 10.4310/JDG/1214437661. URL: https://doi.org/10.4310/JDG/1214437661
5. Werner Ballmann, Misha Brin and Patrick Eberlein, Structure of manifolds of nonpositive curvature. I, Annals of Mathematics (2) 122 (1985), 171-203. DOI: 10.2307/1971373. URL: https://doi.org/10.2307/1971373
6. Viktor Schroeder, A splitting theorem for spaces of nonpositive curvature, Inventiones Mathematicae 79 (1985), 323-327. DOI: 10.1007/BF01388977. URL: https://doi.org/10.1007/BF01388977
7. Werner Ballmann and Patrick Eberlein, Fundamental groups of manifolds of nonpositive curvature, Journal of Differential Geometry 25 (1987), 1-22. DOI: 10.4310/JDG/1214440722. URL: https://doi.org/10.4310/JDG/1214440722
8. Pierre-Emmanuel Caprace and Nicolas Monod, Isometry groups of non-positively curved spaces: discrete subgroups, Journal of Topology 2 (2009), 701-746. DOI: 10.1112/jtopol/jtp027; arXiv:0901.1022. URL: https://doi.org/10.1112/jtopol/jtp027
9. T. Tam Nguyen Phan, On finite volume, negatively curved manifolds, preprint (2011). arXiv:1110.4087. URL: https://arxiv.org/abs/1110.4087
10. Grigori Avramidi, Ends of finite volume, nonpositively curved manifolds, Mathematische Annalen 373 (2019), 191-235. DOI: 10.1007/s00208-018-1698-4; arXiv:1512.01873. URL: https://doi.org/10.1007/s00208-018-1698-4
11. Grigori Avramidi, T. Tam Nguyen Phan and Yunhui Wu, Fundamental groups of finite volume, bounded negatively curved 4-manifolds are not 3-manifold groups, preprint (2013). arXiv:1309.0043. URL: https://arxiv.org/abs/1309.0043
12. Grigori Avramidi and T. Tam Nguyen Phan, Examples of noncompact nonpositively curved manifolds, Bulletin of the London Mathematical Society 53 (2021), 1061-1071. DOI: 10.1112/blms.12484; arXiv:1711.03629. URL: https://doi.org/10.1112/blms.12484
13. Grigori Avramidi and T. Tam Nguyen Phan, Half-dimensional collapse of ends of manifolds of nonpositive curvature, Geometric and Functional Analysis 29 (2019), 1638-1702. DOI: 10.1007/s00039-019-00515-2; arXiv:1608.02185. URL: https://doi.org/10.1007/s00039-019-00515-2
14. Ran Ji and Yunhui Wu, On ends of finite-volume noncompact manifolds of nonpositive curvature, Inventiones mathematicae 237 (2024), 735-777. DOI: 10.1007/s00222-024-01266-0. URL: https://doi.org/10.1007/s00222-024-01266-0
15. James Dibble, Central splitting of manifolds with no conjugate points, Pacific Journal of Mathematics 306 (2020), 95-114. DOI: 10.2140/PJM.2020.306.95. URL: https://doi.org/10.2140/PJM.2020.306.95