arXiv research

Simpler proof for Kielak's virtual fibering criterion.

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over $S^1$ with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive Kodaira dimension, then the fiber itself is a sphere or torus bundle over $S^1$. In a different direction, we prove that if the …

We use virtual knot theory to detect the non-invertibility of some classical links in $\mathbb{S}^3$. These links appear in the study of virtual covers. Briefly, a virtual cover associates a virtual knot $\upsilon$ to a knot $K$ in a $3$-manifold $N$, under certain hypotheses on $K$ and $N$. Virtual covers of links in …

We show that all two-bridge knot and link complements are virtually fibered. We also show that spherical Montesinos knot and link complements are virtually fibered. This is accomplished by showing that such knot complements are finitely covered by great circle link complements.

Paper constructs hyperbolic Coxeter groups that virtually fiber over Z.

We find two new classes of virtually fibered classic Montesinos links of type $\widetilde{SL_2}$.

We mostly determine which closed smooth oriented 4-manifolds fibering over lower dimensional manifolds are virtually symplectic, i.e. finitely covered by symplectic 4-manifolds.

A fibered concordance of knots, introduced by Harer, is a concordance between fibered knots that is well-behaved with respect to the fibrations. We consider semi-fibered concordance of two component ordered links $L=J \sqcup K$ with $J$ fibered. These are concordances that restrict to fibered concordances on the first …

The paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3-manifold, which generalizes an invariant due to Rubinstein and Wang. The subsurface is virtually embedded if and only if the almost fiber part is aspiral, and in this ca…

We prove that an irreducible 3-manifold whose fundamental group satisfies a certain group-theoretic property called RFRS is virtually fibered. As a corollary, we show that 3-dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. These include the Seifert Weber …

In 2007 Agol showed that if N is an aspherical compact 3-manifold with empty or toroidal boundary such that its fundamental group is virtually RFRS, then $N$ is virtually fibered. We give a largely self-contained proof of Agol's theorem using complexities of sutured manifolds.

New trick builds hyperbolic manifolds from compact ones, proving some don't virtually fiber.

Geometric interpretations of some virtual knot invariants are given in terms of invariants of links in $\mathbb{S}^3$. Alexander polynomials of almost classical knots are shown to be specializations of the multi-variable Alexander polynomial of certain two-component boundary links of the form $J \sqcup K$ with $J$ a fi…

Let $p$ be a prime. In this paper, we classify the geometric 3-manifolds whose fundamental groups are virtually residually $p$. Let $M=M^3$ be a virtually fibered 3-manifold. It is well-known that $G=π_1(M)$ is residually solvable and even residually finite solvable. We prove that $G$ is always virtually residually $p$…

Suppose that M is a fibered three-manifold whose fiber is a surface of positive genus with one boundary component. Assume that M is not a semi-bundle. We show that infinitely many fillings of M along dM are virtually Haken. It follows that infinitely many Dehn-surgeries of any non-trivial knot in the three-sphere are v…

New groups found that don't virtually algebraically fiber, related to mapping class group orbits.

The infimal Heegaard gradient of a compact 3-manifold was defined and studied by Marc Lackenby in an approach toward the well-known virtually Haken conjecture. As instructive examples, we consider Seifert fibered 3-manifolds, and show that a Seifert fibered 3-manifold has zero infimal Heegaard gradient if and only if i…

We prove that if a closed oriented 4-manifold X fibers over a 2- or 3-dimensional manifold, in most cases all of its virtual Betti numbers are infinite. In turn, we show that a closed oriented 4-manifold X which is not a tower of torus bundles and fibering over a 2- or 3-dimensional manifold does not admit a torsion sy…

We succeed to generalize spun knots of classical 1-knots to the virtual 1-knot case by using the `spinning construction'. That, is, we prove the following: Let $Q$ be a spun knot of a virtual 1-knot $K$ by our method. The embedding type $Q$ in $S^4$ depends only on $K$. Furthermore we prove the following: The submanifo…

Proves incoherence of free-by-free and surface-by-free groups, solving two problems.

In recent years, the RFRS condition has been used to analyze virtual fibering in 3-manifold topology. Agol's work shows that any 3-manifold with zero Euler characteristic satisfying the RFRS condition on its fundamental group virtually fibers over the circle. In this note we will show that a finitely generated nilpoten…

We introduce a new technique for studying classical knots with the methods of virtual knot theory. Let $K$ be a knot and $J$ a knot in the complement of $K$ with $\text{lk}(J,K)=0$. Suppose there is covering space $π_J: Σ\times (0,1) \to \bar{S^3\backslash V(J)}$, where $V(J)$ is a regular neighborhood of $J$ satisfyin…

Study shows hyperbolic subgroups can be free products of surface and free groups.

We show that if a closed hyperbolic 3-manifold has infinitely many finite covers of bounded Heegaard genus, then it is virtually fibered. This generalizes a theorem of Lackenby, removing restrictions needed about the regularity of the covers. Furthermore, we can replace the assumption that the covers have bounded Heega…

The abstract discusses conjectures about virtual Legendrian knots and their relation to causality.

In this note we prove that any closed graph manifold admitting a metric of non-positive sectional curvature (NPC-metric) has a finite cover, which is fibered over the circle. An explicit criterion to have a finite cover, which is fibered over the circle, is presented for the graph manifolds of certain class.

We use Heegaard splittings to give a criterion for a tunnel number one knot manifold to be non-fibered and to have large cyclic covers. We also show that such a knot manifold (satisfying the criterion) admits infinitely many virtually Haken Dehn fillings. Using a computer, we apply this criterion to the 2 generator, no…

We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are separable. A consequence is that closed hyperbolic 3-manifolds have finite-sheeted Haken covers, which resolves the virtual Haken …

In this paper we generalize the notion of strongly poly-free group to a larger class of groups, we call them strongly poly-surface groups and prove that the Fibered Isomorphism Conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for any virtually strongly poly-surface g…

We consider the following properties of compact oriented irreducible graph-manifolds: to contain a $π_1$-injective surface (immersed, virtually embedded or embedded), be (virtually) fibered over $S^1$, and to carry a metric of nonpositive sectional curvature. It turns out that all these properties can be described from…

We show that every closed, virtually fibered hyperbolic 3-manifold contains immersed, quasi-Fuchsian surfaces with convex cores of arbitrarily large thickness.

It is known that the lens space $L(2n,1)$ supports a virtually overtwisted contact structure arising as the boundary of the Milnor fiber of a complex hypersurface singularity. In this article we study the problem of realizing other $(L(p,q),ξ)$ in such a way, obtaining a series of necessary conditions for this to happe…

Study symplectic fillings of lens spaces, focusing on virtually overtwisted contact structures.

We show that if M is a surface bundle over S^1 with fiber of genus 2, then for any integer n, M has a finite cover tilde(M) with b_1(tilde(M)) > n. A corollary is that M can be geometrized using only the `non-fiber' case of Thurston's Geometrization Theorem for Haken manifolds.

Proves hyperbolized groups are virtually compact special and linear.

This paper stems from the observation (arising from work of T. Delzant) that "most" Kähler groups virtually algebraically fiber, i.e. admit a finite index subgroup that maps onto $\Bbb{Z}$ with finitely generated kernel. For the remaining ones, the Albanese dimension of all finite index subgroups is at most one, i.e. t…

Proof of existence for ideal triangulations that normalize fibers in certain 3-manifolds.

The Farrell-Jones Fibered Isomorphism Conjecture for the stable topological pseudoisotopy theory has been proved for several classes of groups. For example for discrete subgroups of Lie groups, virtually poly-infinite cyclic groups, Artin braid groups, a class of virtually poly-surface groups and virtually solvable lin…

The study shows that certain Artin groups cannot contain hyperbolic manifold groups.

Let M be a complete hyperbolic 3-manifold of finite volume that admits a decomposition into right-angled ideal polyhedra. We show that M has a deformation retraction that is a virtually special square complex, in the sense of Haglund and Wise and deduce that such manifolds are virtually fibered. We generalise a theorem…

We present a simulation framework for spunbond processes and use a design of experiments to investigate the cause-and-effect-relations of process and material parameters onto the fiber laydown on a conveyor belt. The virtual experiments are analyzed by a blocked neural network. This forms the basis for the prediction o…

We define a category $v\mathcal{T}$ of tangles diagrams drawn on surfaces with boundaries. On the one hand we show that there is a natural functor from the category of virtual tangles to $v\mathcal{T}$ which induces an equivalence of categories. On the other hand, we show that $v\mathcal{T}$ is universal among ribbon c…

The paper proves drilled bundles over graphs are virtually special cubulable.

The derived group of a permutation representation, introduced by R.H. Crowell, unites many notions of knot theory. We survey Crowell's construction, and offer new applications. The twisted Alexander group of a knot is defined. Using it, we obtain twisted Alexander modules and polynomials. Also, we extend a well-known t…

A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe computer experiments whic…

The purpose of this paper is to produce restrictions on fundamental groups of manifolds admitting good complexifications by proving the following Cheeger-Gromoll type splitting theorem: Any closed manifold $M$ admitting a good complexification has a finite-sheeted regular covering $M_1$ such that $M_1$ admits a fiber b…

Study $L^2$-Betti numbers of Dehn fillings for special groups.

Suppose M is a cusped finite-volume hyperbolic 3-manifold and T is an ideal triangulation of M with essential edges. We show that any incompressible surface S in M that is not a virtual fiber can be isotoped into spunnormal form in T . The proof is based directly on ideas of W. Thurston.