arXiv research

The study classifies and explores $SU(2)$-cyclic and $SU(2)$-abelian 3-manifolds.

Computes virtually cyclic dimension for 3-manifold groups.

Smooth actions of cyclic groups on 3-manifolds are conjugate to smooth ones.

3-manifolds study Hasse norm principle, akin to number fields.

New 3-manifold spines with unique Whitehead graphs identified.

Study on SU(2)-simple knots and cyclic 3-manifolds, extending known results.

We show that a hyperbolic $3$-manifold can be the cyclic branched cover of at most fifteen knots in $\mathbf{S}^3$. This is a consequence of a general result about finite groups of orientation preserving diffeomorphisms acting on $3$-manifolds. A similar, although weaker, result holds for arbitrary irreducible $3$-mani…

Study proves non-left-orderability of 3-manifolds derived from specific knots.

We show that the infinite cyclic cover of the exterior of the untwisted Whitehead double of a non-trivial knot does not embed in any compact 3-manifold, answering a question of Jiang, Ni, Wang and Zhou.

We show that if the lower central series of the fundamental group of a closed oriented $3$-manifold stabilizes then the maximal nilpotent quotient is a cyclic group, a quaternion $2$-group cross an odd order cyclic group, or a Heisenberg group. These groups are well known to be precisely the nilpotent fundamental group…

The paper explores circular orderability in 3-manifold groups, related to the L-space conjecture.

In this article we give combinatorial criteria to decide whether a transitive cyclic combinatorial d-manifold can be generalized to an infinite family of such complexes, together with an explicit construction in the case that such a family exists. In addition, we substantially extend the classification of combinatorial…

We are interested in finite groups acting orientation-preservingly on 3-manifolds (arbitrary actions, ie not necessarily free actions). In particular we consider finite groups which contain an involution with nonempty connected fixed point set. This condition is satisfied by the isometry group of any hyperbolic cyclic …

We study the cyclic presentations with relators of the form $x_ix_{i+m}x_{i+k}^{-1}$ and the groups they define. These "groups of Fibonacci type" were introduced by Johnson and Mawdesley and they generalize the Fibonacci groups $F(2,n)$ and the Sieradski groups $S(2,n)$. With the exception of two groups, we classify wh…

We consider the discrete representations of 3-manifold groups into $PU(2,1)$ that appear in the Falbel-Koseleff-Rouillier census, such that the peripheral subgroups have cyclic unipotent holonomy. We show that two of these representations have conjugate images, even though they represent different 3-manifold groups. Th…

Toroidal 3-manifolds have special group structures that can be shown through specific covers.

The paper expands Dunwoody's classification of graphs to include all graphs satisfying the first five conditions.

The paper presents a chain complex for 3-manifold covers, including surface bundles and surgeries.

We show that a random 3-manifold with positive first Betti number admits a tower of cyclic covers with exponential torsion growth.

This paper uses Brin and Thickstun's theory of end reductions of non-compact 3-manifolds to study groups of covering translations of irreducible contractible open 3-manifolds W which are not homeomorphic to R^3. We associate to W an object S(W) called the simplicial complex of minimal R^2-irreducible end reductions of …

Updated rerefences and introduction. Given a knot in an integer homology sphere, one can construct a family of closed 3-manifolds (parametrized by the positive integers), namely the cyclic branched coverings of the knot. In this paper we give a formula for the the Casson-Walker invariants of these 3-manifolds in terms …

New quantum invariant is asymptotically multiplicative under cyclic covers.

We discuss 3-manifolds which are cyclic coverings of the 3-sphere, branched over 2-bridge knots and links. Different descriptions of these manifolds are presented: polyhedral, Heegaard diagram, Dehn surgery and coloured graph constructions. Using these descriptions, we give presentations for their fundamental groups, w…

In this paper, we prove a geometrization conjecture, every orientable smooth closed 3-manifold with finite fundamental group is homeomorphic to $S^3/G$ for some finite cyclic subgroup $G\subset {Isom}^+(S^3)$.

In this paper we show that periodic Takahashi 3-manifolds are cyclic coverings of the connected sum of two lens spaces (possibly cyclic coverings of the 3-sphere), branched over knots. When the base space is a 3-sphere, we prove that the associated branching set is a two-bridge knot of genus one, and we determine its t…

In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually $\bf S^3$), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of …

Study shows non-cyclic groups of diffeomorphisms can't act on certain 3-manifolds.

Let $M$ be an irreducible, compact, connected, orientable 3-manifold whose boundary is a torus. We show that if $M$ is hyperbolic, then it admits at most six finite/cyclic fillings of maximal distance 5. Further, the distance of a finite/cyclic filling to a cyclic filling is at most 2. If $M$ has a non-boundary-paralle…

We prove that fundamental groups of non-orientable 3-manifolds have a solvable conjugacy problem, and construct an algorithm. Together with our earlier work on the conjugacy problem in groups on orientable geometrizable 3-manifolds, all $π_1$ of (geometrizable) 3-manifolds have a solvable conjugacy problem. In corollar…

Study of $η$-Ricci solitons on Kenmotsu 3-manifolds.

We study the dependence of solutions of equations of the form $a_0 + a_1 z^{\ell_1} + ... + a_m z^{\ell_m} = 0$, on the exponents $\ell_1, ..., \ell_m$. We apply our results to equations that appear in graph theory, the theory of 3-manifolds fibering over the circle, and the theory of free-by-cyclic groups. In particul…

Given a real analytic function $f$ from $\mathbb{R}^4$ to $\mathbb{R}^2$ with isolated critical point at the origin, the link $L_f$ of the singularity is a real fibred knot in $\mathbb{S}^{3}$. From this singularities, we construct a family of real isolated suspension singularities from $\mathbb{R}^6$ to $\mathbb{R}^2$…

Given a 3-manifold M with no spherical boundary components, and a primitive class φin H^1(M;Z), we show that the following are equivalent: (1) φis a fibered class, (2) the rank gradient of (M,φ) is zero, (3) the Heegaard gradient of (M,φ) is zero.

From the homotopy groups of two cubic spherical 3-manifolds we construct the isomorphic groups of deck transformations acting on the 3-sphere. These groups become the cyclic group of order eight and the quaternion group respectively. By reduction of representations from the orthogonal group to the identity representati…

Study shows universal circles for taut foliations in 3-manifolds.

We say a knot $k$ in the 3-sphere $\mathbb S^3$ has {\it Property $IE$} if the infinite cyclic cover of the knot exterior embeds into $\mathbb S^3$. Clearly all fibred knots have Property $IE$. There are infinitely many non-fibred knots with Property $IE$ and infinitely many non-fibred knots without property $IE$. Both…

Study on pretzel knots and their left-orderable properties.

Study algebraic K-theory of 3-manifold groups using Farrell-Jones isomorphism and geometrization.

We consider irreducible 3-manifolds M that arise as knot complements in closed 3-manifolds and that contain at most two connected strict essential surfaces. The results in the paper relate the boundary slopes of the two surfaces to their genera and numbers of boundary components. Explicit quantitative relationships, wi…

In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of 3-manifolds with transitive cyclic symmetry can be generalised to an infinite fami…

Formulates Hilbert reciprocity law on 3-manifolds.

The renormalized volume is reinterpreted using isoperimetric profiles.

Study on realizing subgroup twists in 3-manifolds.

In this paper it is proven that if the group of covering translations of the covering space of a compact, connected, $P^2$-irreducible 3-manifold corresponding to a non-trivial, finitely-generated subgroup of its fundamental group is infinite, then either the covering space is almost compact or the subgroup is infinite…

Detect slopes in toroidal 3-manifolds to prove properties of fundamental groups.

Given an irreducible contractible open 3-manifold W which is not homeomorphic to R^3, there is an associated simplicial complex S(W), the complex of end reductions of W. Whenever W covers a 3-manifold M one has that the fundamental group of M is isomorphic to a subgroup of the group Aut(S(W)) of simplicial automorphism…

We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched covers of S^3 branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable.…

Compact actions on manifolds with specific eigenvalue conditions.