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.

Time-Dependent Lagrangian Energy Systems on Supermanifolds with Graph Bundles

The aim of this article is firstly to improve time-dependent Lagrangian energy equations using the super jet bundles on supermanifolds. Later, we adapted this study to the graph bundle. Thus, we created a graph bundle by examining the graph manifold structure in superspace. The geometric structures obtained for the mechanical energy system with superbundle coordinates were reexamined with the graph bundle coordinates. Thus, we were able to calculate the energy that occurs during the motion of a particle when we examine this motion with graph points. The supercoordinates on the superbundle structure of supermanifolds have been given for body and soul and also even and odd dimensions. We have given the geometric interpretation of this property in coordinates for the movement on graph points. Lagrangian energy equations have been applied to the presented example, and the advantage of examining the movement with graph points was presented. In this article, we will use the graph theory to determine the optimal motion, velocity, and energy of the particle, due to graph points. This study showed a physical application and interpretation of supervelocity and supertime dimensions in super-Lagrangian energy equations utilizing graph theory.

Constructing Dynamic Brain Functional Networks via Hyper-Graph Manifold Regularization for Mild Cognitive Impairment Classification

Brain functional networks (BFNs) constructed via manifold regularization (MR) have emerged as a powerful tool in finding new biomarkers for brain disease diagnosis. However, they only describe the pair-wise relationship between two brain regions, and cannot describe the functional interaction between multiple brain regions, or the high-order relationship, well. To solve this issue, we propose a method to construct dynamic BFNs (DBFNs) via hyper-graph MR (HMR) and employ it to classify mild cognitive impairment (MCI) subjects. First, we construct DBFNs via Pearson’s correlation (PC) method and remodel the PC method as an optimization model. Then, we use k-nearest neighbor (KNN) algorithm to construct the hyper-graph and obtain the hyper-graph manifold regularizer based on the hyper-graph. We introduce the hyper-graph manifold regularizer and the L1-norm regularizer into the PC-based optimization model to optimize DBFNs and obtain the final sparse DBFNs (SDBFNs). Finally, we conduct classification experiments to classify MCI subjects from normal subjects to verify the effectiveness of our method. Experimental results show that the proposed method achieves better classification performance compared with other state-of-the-art methods, and the classification accuracy (ACC), the sensitivity (SEN), the specificity (SPE), and the area under the curve (AUC) reach 82.4946 ± 0.2827%, 77.2473 ± 0.5747%, 87.7419 ± 0.2286%, and 0.9021 ± 0.0007, respectively. This method expands the MR method and DBFNs with more biological significance. It can effectively improve the classification performance of DBFNs for MCI, and has certain reference value for the research and auxiliary diagnosis of Alzheimer’s disease (AD).

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.

Quasi-isometries of pairs: Surfaces in graph manifolds

We show there exists a closed graph manifold [Formula: see text] and infinitely many non-separable, horizontal surfaces [Formula: see text] such that there does not exist a quasi-isometry [Formula: see text] taking [Formula: see text] to [Formula: see text] within a finite Hausdorff distance when [Formula: see text].

Contracting elements and random walks

We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper {\mathrm{CAT}(0)} spaces, elements acting hyperbolically on the Bass–Serre tree in graph manifold groups. We also define a related notion of weakly contracting element, and show that those coincide with hyperbolic elements in groups acting acylindrically on hyperbolic spaces and with iwips in {\mathrm{Out}(F_{n})} , {n\geq 3} . We show that each weakly contracting element is contained in a hyperbolically embedded elementary subgroup, which allows us to answer a problem in [16]. We prove that any simple random walk in a non-elementary finitely generated subgroup containing a (weakly) contracting element ends up in a non-(weakly-)contracting element with exponentially decaying probability.