arXiv research

New proof limits Jones polynomial values for quasi-alternating links.

Although there are infinitely many knots with superbridge index n for every even integer n>2, there are only finitely many knots with superbridge index 3.

Finitely many pseudo-Anosov flows without perfect fits in a 3-manifold.

Many 2D Artin groups are residually finite.

We show that 0.29 is a Margulis number for all but finitely many hyperbolic 3-manifolds. The finitely many exceptions are all closed.

In this paper, we proved that any closed orientable 3-manifold 1-dominates at most finitely many geometric 3-manifolds.

We construct a finitely presented group with infinitely many non-homeomorphic asymptotic cones. We also show that the existence of cut points in asymptotic cones of finitely presented groups does, in general, depend on the choice of scaling constants and ultrafilters.

Study finds many Lagrangian fillings for certain Legendrian links.

A countable CW complex $K$ is quasi-finite (as defined by A.Karasev) if for every finite subcomplex $M$ of $K$ there is a finite subcomplex $e(M)$ such that any map $f:A\to M$, where $A$ is closed in a separable metric space $X$ satisfying $XτK$, has an extension $g:X\to e(M)$. Levin's results imply that none of the Ei…

Accessible groups with infinitely many ends have infinitely many twisted conjugacy classes.

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference "Finite-dimensional Integrable Systems, FDIS 2017" held at CRM, Barcelona in July 2017.

We show that there are at most finitely many one cusped orientable hyperbolic 3-manifolds which have more than eight non-hyperbolic Dehn fillings. Moreover, we show that determining these finitely many manifolds is decidable.

Curves can bound only finitely many developable surfaces.

In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We give uncountably many groups of type $FP_2$ with similar properties to those subgro…

It is a consequence of the classical Jordan bound for finite subgroups of linear groups that in each dimension n there are only finitely many finite simple groups which admit a faithful, linear action on the n-sphere. In the present paper we prove an analogue for smooth actions on arbitrary homology n-spheres: in each …

We construct Kähler groups with arbitrary finiteness properties by mapping products of closed Riemann surfaces holomorphically onto an elliptic curve: for each $r\geq 3$, we obtain large classes of Kähler groups that have classifying spaces with finite $(r-1)$-skeleton but do not have classifying spaces with finitely m…

The study shows that symplectic Lefschetz fibrations can have infinitely many sections.

In this paper, it is shown that every orientable closed 3-manifold maps with nonzero degree onto at most finitely many homeomorphically distinct irreducible non-geometric orientable closed 3-manifolds. Moreover, given any nonzero integer, as a mapping degree up to sign, every orientable closed 3-manifold maps with that…

The study finds only finitely many Kähler-Einstein compactifications for semisimple groups.

We show that up to commensurability there are only finitely many cocompact arithmetic Kleinian groups generated by rotations. This implies, in particular, that there exist only finitely many conjugacy classes of cocompact two generated arithmetic Kleinian groups. The proof of the main result is based on a generalized G…

Finite totally geodesic hypersurfaces in curved manifolds proven.

We prove that there are only finitely many arithmetic Kleinian maximal reflection groups.

Research shows finiteness in triangulations with girth constraints.

We show that large classes of non-arithmetic hyperbolic $n$-manifolds, including the hybrids introduced by Gromov and Piatetski-Shapiro and many of their generalizations, have only finitely many finite-volume immersed totally geodesic hypersurfaces. In higher codimension, we prove finiteness for geodesic submanifolds o…

We show that many spin 6-manifolds have the homotopy type but not the homeomorphism type of a Kaehler manifold. Moreover, for given Betti numbers, there are only finitely many deformation types and hence topological types of smooth complex projectve spin threefolds of general type. Finally, on a fixed spin 6-manifold, …

We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.

We prove a finiteness result for the $\partial$-patterned guts decomposition of all 3-manifolds obtained by splitting a given orientable, irreducible and $\partial$-irreducible 3-manifold along a closed incompressible surface. Then using the Thurston norm, we deduce that the JSJ-pieces of all 3-manifolds dominated by a…

The paper explores the structure of Reeb spaces for smooth functions on manifolds.

In this article we collect a series of observations that constrain actions of many groups on compact manifolds. In particular, we show that "generic" finitely generated groups have no smooth volume preserving actions on compact manifolds while also producing many finitely presented, torsion free groups with the same pr…

We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.

We construct infinitely many noncommensurable non-cocompact Fuchsian groups $Δ$ of finite covolume sitting in PSL(2,Q) so that the set of hyperbolic fixed points of $Δ$ will contain a given finite collection of elements in the boundary of the hyperbolic plane.

We investigate the curvature of invariant metrics on G-manifolds with finitely many non-principal orbits. We prove existence results for metrics of positive Ricci curvature and non-negative sectional curvature, and discuss some families of examples to which these existence results apply.

A maximal surface $\sb$ with isolated singularities in a complete flat Lorentzian 3-manifold $\N$ is said to be entire if it lifts to a (periodic) entire multigraph $\tilde{\sb}$ in $ł^3.$ In addition, $\sb$ is called of finite type if it has finite topology, finitely many singular points and $\tilde{\sb}$ is finitely …

Consider a group G and a family $\mathcal{A}$ of subgroups of G. We say that vertex finiteness holds for splittings of G over $\mathcal{A}$ if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal G-trees with edge stabilizers in $\mathcal{A}$. We show vertex finiteness when G…

The study of orthospectrum and simple orthospectrum of hyperbolic surfaces.

We consider manifolds $M^{2n}$ which admit smooth maps into a connected sum of $S^1\times S^n$ with only finitely many critical points, for $n\in\{2,4,8\}$, and compute the minimal number of critical points.

Study of uncountable family of finitely generated groups.

We show that only finitely many links in a closed 3-manifold share the same complement, up to twists along discs and annuli. Using the same techniques, we prove that by adding 2-handles on the same link we get only finitely many smooth cobordisms between two given closed 3-manifolds. As a consequence, there are finitel…

New groups algebraically fibre with high-dimensional hyperbolic groups.

This paper adresses the following problem: Given a closed orientable three-manifold M, are there at most finitely many closed orientable three-manifolds 1-dominated by M? We solve this question for the class of closed orientable graph manifolds. More presisely the main result of this paper asserts that any closed orien…

In the present paper, we show that many combinatorial and topological objects, such as maps, hypermaps, three-dimensional pavings, constellations and branched coverings of the two--sphere admit any given finite automorphism group. This enhances the already known results by Frucht, Cori -- Machì, Širáň -- Škoviera, and …

For most positive integer pairs $(a,b)$, the topological space $#a{\mathbb C \mathbb P}^2#b{\bar{\mathbb C \mathbb P^2}}$ is shown to admit infinitely many inequivalent smooth structures which dissolve upon performing a single connected sum with $S^2\times S^2$. This is then used to construct infinitely many non-equiva…

We study aspherical manifolds that do not support Anosov diffeomorphisms. Weakening conditions of Gogolev and Lafont, we show that the product of an infranilmanifold with finitely many aspherical manifolds whose fundamental groups have trivial center and finite outer automorphism group does not support Anosov diffeomor…

Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.

The existence of essential closed surfaces surfaces is proven for finite coverings of 3-manifolds that are triangulated by finitely many topological ideal tetrahedra and admit a regular, negatively curved, ideal structure.

We consider the ortho spectrum of hyperbolic surfaces with totally geodesic boundary. We show that in general the ortho spectrum does not determine the systolic length but that there are only finitely many possibilities. As a corollary we show that, up to isometry, there are only finitely many hyperbolic structures on …

New result on finite gerbes simplifies classification of certain bundles.

New groups found in hyperbolic space with infinite fields of definition.