Graph Manifolds Research Articles

Volume of Seifert representations for graph manifolds and their finite covers

AbstractFor any closed orientable 3‐manifold, there is a volume function defined on the space of all Seifert representations of the fundamental group. The maximum absolute value of this function agrees with the Seifert volume of the manifold due to Brooks and Goldman. For any Seifert representation of a graph manifold, the authors establish an effective formula for computing its volume, and obtain restrictions to the representation as analogous to the Milnor–Wood inequality (about transversely projective foliations on Seifert fiber spaces). It is shown that the Seifert volume of any graph manifold is a rational multiple of . Among all finite covers of a given nongeometric graph manifold, the supremum ratio of the Seifert volume over the covering degree can be a positive number, and can be infinite. Examples of both possibilities are discovered, and confirmed with the explicit values determined for the finite ones.

Bott‐integrable Reeb flows on 3‐manifolds

AbstractThis paper is devoted to studying a notion of Bott integrability for Reeb flows on contact 3‐manifolds. We show, in analogy with work of Fomenko–Zieschang on Hamiltonian flows in dimension 4, that Bott‐integrable Reeb flows exist precisely on graph manifolds. We also show that all ‐invariant contact structures on Seifert manifolds, as well as all contact structures on the 3‐sphere, on the 3‐torus and on , admit Bott‐integrable Reeb flows. Along the way, we establish some general Liouville‐type theorems for Bott‐integrable Reeb flows, and a number of topological constructions (connected sum, open books, Dehn surgery) that may be expected to have wider applications.

Croke–Kleiner Admissible Groups: Property (QT) and Quasiconvexity

Croke–Kleiner admissible groups first introduced by Croke and Kleiner [CK02] belong to a particular class of graphs of groups, which generalizes fundamental groups of three-dimensional graph manifolds. In this paper, we show that if G is a Croke–Kleiner admissible group, then a finitely generated subgroup of G has finite height if and only if it is strongly quasiconvex. We also show that if G↷X is a flip CKA action, then G is quasiisometrically embedded into a finite product of quasitrees. With further assumption on the vertex groups of the flip CKA action G↷X, we show that G satisfies property (QT) introduced by Bestvina, Bromberg, and Fujiwara [BBF21].

Virtually unipotent curves in some non-NPC graph manifolds

Let [Formula: see text] be a graph manifold containing a single JSJ torus [Formula: see text] and whose JSJ blocks are of the form [Formula: see text], where [Formula: see text] is a compact orientable surface with boundary. We show that if [Formula: see text] does not admit a Riemannian metric of everywhere nonpositive sectional curvature, then there is an essential curve on [Formula: see text] such that any finite-dimensional linear representation of [Formula: see text] maps an element representing that curve to a matrix all of whose eigenvalues are roots of [Formula: see text]. In particular, this shows that [Formula: see text] does not admit a faithful finite-dimensional unitary representation, and gives a new proof that [Formula: see text] is not linear over any field of positive characteristic.

Spines and surgery descriptions of graph manifolds

Graph manifolds are compact orientable 3–manifolds obtained by gluing several copies of D2×S1 and N2×S1 together by homeomorphisms of some components of their boundaries (D2 is the 2–disc and N2 denotes the 2–disc with two holes). Here we study spines and surgery representations of orientable graph manifolds, and derive geometric presentations of their fundamental group. Then we determine the homeomorphism type of many Takahashi manifolds and the Teragaito manifolds, showing that they are graph manifolds with specified invariants. Finally, we describe graph manifolds arising from toroidal surgeries on certain classes of hyperbolic knots.

Complete topological descriptions of certain Morse boundaries

We study direct limits of embedded Cantor sets and embedded Sierpiński curves. We show that under appropriate conditions on the embeddings, all limits of Cantor spaces give rise to homeomorphic spaces, called \omega -Cantor spaces, and, similarly, all limits of Sierpiński curves give homeomorphic spaces, called \omega -Sierpiński curves. We then show that the former occur naturally as Morse boundaries of right-angled Artin groups and fundamental groups of non-geometric graph manifolds, while the latter occur as Morse boundaries of fundamental groups of finite-volume, cusped hyperbolic 3-manifolds.

The topology of Bott integrable fluids

<p style='text-indent:20px;'>We construct non-vanishing steady solutions to the Euler equations (for some metric) with analytic Bernoulli function in each three-manifold where they can exist: graph manifolds. Using the theory of integrable systems, any admissible Morse-Bott function can be realized as the Bernoulli function of some non-vanishing steady Euler flow. This can be interpreted as an inverse problem to Arnold's structure theorem and yields as a corollary the topological classification of such solutions. Finally, we prove that the topological obstruction holds without the non-vanishing assumption: steady Euler flows with a Morse-Bott Bernoulli function only exist on graph three-manifolds.</p>

Residual finiteness and Abelian subgroup separability of some high dimensional graph manifolds

We generalize $3$-manifolds supporting non-positively curved metric to construct manifolds which have the following properties : (1) They are not locally $\mathrm{CAT}(0)$. (2) Their fundamental groups are residually finite. (3) They have subgroup separability for some abelian subgroups.

Entropy and finiteness of groups with acylindrical splittings

We prove that there exists a positive, explicit function F(k, E) such that, for any group G admitting a k -acylindrical splitting and any generating set S of G with \operatorname{Ent}(G,S) , we have |S| \leq F(k, E) . We deduce corresponding finiteness results for classes of groups possessing acylindrical splittings and acting geometrically with bounded entropy: for instance, D -quasiconvex k -malnormal amalgamated products acting on \delta -hyperbolic spaces or on \operatorname{CAT}(0) -spaces with entropy bounded by E . A number of finiteness results for interesting families of Riemannian or metric spaces with bounded entropy and diameter also follow: Riemannian 2-orbifolds, non-geometric 3 -manifolds, higher dimensional graph manifolds and cusp-decomposable manifolds, ramified coverings and, more generally, \operatorname{CAT}(0) -groups with negatively curved splittings.

Conjugacy Problem in the Fundamental Groups of High-Dimensional Graph Manifolds

The conjugacy problem for a group G is one of the important algorithmic problems deciding whether or not two elements in G are conjugate to each other. In this paper, we analyze the graph of group structure for the fundamental group of a high-dimensional graph manifold and study the conjugacy problem. We also provide a new proof for the solvable word problem.

Geometric graph manifolds with non‐negative scalar curvature

We classify $n$-dimensional geometric graph manifolds with nonnegative scalar curvature, and first show that if $n>3$, the universal cover splits off a codimension 3 Euclidean factor. We then proceed with the classification of the 3-dimensional case by showing that such a manifold is either a lens space or a prism manifold with a very rigid metric. This allows us to also classify the moduli space of such metrics: it has infinitely many connected components for lens spaces, while it is connected for prism manifolds.

Algebraic ranks of the fundamental groups of high dimensional graph manifolds

Algebraic ranks of the fundamental groups of high dimensional graph manifolds

Graph manifolds as ends of negatively curved Riemannian manifolds

Let $M$ be a graph manifold such that each piece of its JSJ decomposition has the $\Bbb H^2 \times \Bbb R$ geometry. Assume that the pieces are glued by isometries. Then, there exists a complete Riemannian metric on $\Bbb R \times M$ which is an "eventually warped cusp metric" with the sectional curvature $K$ satisfying $-1 \le K <0$. A theorem by Ontaneda then implies that $M$ appears as an end of a 4-dimensional, complete, non-compact Riemannian manifold of finite volume with sectional curvature $K$ satisfying $-1 \le K <0$.

Rational L‐space surgeries on satellites by algebraic links

Given an $n$-component link $L$ in any 3-manifold $M$, the space $\mathcal{L} \subset (\mathbb{Q}\cup \mkern-1.5mu\{\infty\})^n$ of rational surgery slopes yielding L-spaces is already fully characterized (in joint work by the author) when $n\!=\!1$ and $\mathcal{L}$ is nontrivial. For $n\mkern-2mu>\mkern-3mu1$, however, there are no previous results for $\mathcal{L}$ as a rational subspace, and only limited results for integer surgeries $\mathcal{L}\cap\mathbb{Z}^n$ on $S^3\mkern-2mu$. Herein, we provide the first nontrivial explicit descriptions of $\mathcal{L}$ for rational surgeries on multi-component links. Generalizing Hedden's and Hom's L-space result for cables, we compute both $\mathcal{L}$, and its topology, for all satellites by torus-links in $S^3\mkern-2mu$. For fractal-boundaried $\mathcal{L}$ resulting from satellites by algebraic links or iterated torus links, we develop arbitrarily precise approximation tools. We also extend the provisional validity of the L-space conjecture for rational surgeries on a knot $K \subset S^3$ to rational surgeries on such satellite-links of $K$. These results exploit the author's generalized Jankins-Neumann formula for graph manifolds.

Brain graph super-resolution for boosting neurological disorder diagnosis using unsupervised multi-topology connectional brain template learning.

Existing graph analysis techniques generally focus on decreasing the dimensionality of graph data (i.e., removing nodes, edges, or both) in diverse predictive learning tasks in pattern recognition, computer vision, and medical data analysis such as dimensionality reduction, filtering and embedding techniques. However, graph super-resolution is strikingly lacking, i.e., the concept of super-resolving low-resolution (LR) graphs with nr nodes into high-resolution graphs (HR) with [Formula: see text] nodes. Particularly, learning how to automatically generate HR brain connectomes, without resorting to the computationally expensive MRI processing steps such as image registration and parcellation, remains unexplored. To fill this gap, we propose the first technique to super-resolve undirected fully connected graphs with application to brain connectomes. First, we root our brain graph super-resolution (BGSR) framework in learning how to estimate a centered LR population-based brain graph representation, coined as connectional brain template (CBT), acting as a proxy in the target BGSR task. Specifically, we hypothesize that the estimation of a well-representative and centered CBT would help better capture the individuality of each LR brain graph via its residual distance from the population-based CBT. This will eventually allow an accurate identification of the most similar individual graphs to a new testing graph in the LR domain for the target prediction task. Second, we leverage the estimated LR CBT (i.e., population mean) to derive residual LR brain graphs, capturing the deviation of all subjects from the estimated CBT. Third, we learn multi-topology LR graph manifolds using different graph topological measurements (e.g., degree, closeness, betweenness) by estimating residual LR similarity matrices modeling the relationship between pairs of residual LR graphs. These are then fused so we can effectively identify for each testing LR subject its most K similar training LR graphs. Last, the missing testing HR graph is predicted by averaging the HR graphs of the K selected training subjects. Predicted HR from LR functional brain graphs boosted classification results for autistic subjects by 16.48% compared with LR functional graphs.

L-spaces, taut foliations, and graph manifolds

If $Y$ is a closed orientable graph manifold, we show that $Y$ admits a coorientable taut foliation if and only if $Y$ is not an L-space. Combined with previous work of Boyer and Clay, this implies that $Y$ is an L-space if and only if $\unicode[STIX]{x1D70B}_{1}(Y)$ is not left-orderable.

Maniflods with conullity at most two as graph manifolds

Maniflods with conullity at most two as graph manifolds

Characteristic varieties of graph manifolds and quasi-projectivity of fundamental groups of algebraic links

The present paper studies the structure of characteristic varieties of fundamental groups of graph manifolds. As a consequence, a simple proof solving a question posed by Papadima on the characterization of algebraic links that have quasi-projective fundamental groups is provided. The type of quasi-projective obstructions used here are in the spirit of Papadima’s original work.

Profinite rigidity of graph manifolds, II: knots and mapping classes

In this paper we study some consequences of the author’s classification of graph manifolds by their profinite fundamental groups. In particular we study commensurability, the behaviour of knots, and relation to mapping classes. We prove that the exteriors of graph knots are distinguished among all 3-manifold groups by their profinite fundamental groups. We also prove a strong conjugacy separability result for certain mapping classes of surfaces.

Integral approximation of simplicial volume of graph manifolds

Graph manifolds are manifolds that decompose along tori into pieces with a tame $S^1$-structure. In this paper, we prove that the simplicial volume of graph manifolds (which is known to be zero) can be approximated by integral simplicial volumes of their finite coverings. This gives a uniform proof of the vanishing of rank gradients, Betti number gradients and torsion homology gradients for graph manifolds.