arXiv research

The paper finds linked periodic orbits in disc homeomorphisms using braids.

Study strong Nielsen equivalence on punctured disc homeomorphisms.

The paper proves bounded cohomology properties of Euclidean space groups.

This paper extends braid lifting to coloured braid groupoids for all simple disc covers.

Study cohomology of homeomorphisms and diffeomorphisms of manifolds.

Given a proper map f : M $\rightarrow$ Q, having cell-like point-inverses, from a manifold-without-boundary M onto an ANR Q, it is a much-studied problem to find when f is approximable by homeomorphisms, i.e., when the decomposition of M induced by f is shrinkable (in the sense of Bing). If dimension M $\geq$ 5, J. W. …

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal, is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this r…

We prove a reverse isoperimetric inequality for domains homeomorphic to a disc with the boundary of curvature bounded below lying in two-dimensional Alexandrov spaces of curvature $\geqslant c$. We also study the equality case.

If M is a hyperbolic 3-manifold with a quasigeodesic flow then we show that π_1(M) acts in a natural way on a closed disc by homeomorphisms. Consequently, such a flow either has a closed orbit or the action on the boundary circle is Möbius-like but not conjugate into PSL(2, R). We conjecture that the latter possibility…

The Teichmueller polynomial of a fibered 3-manifold plays a useful role in the construction of mapping class having small stretch factor. We provide an algorithm that computes this polynomial of the fibered face associated to a pseudo-Anosov mapping class of a disc homeomorphism. As a byproduct, our algorithm allows us…

In the first part of the paper we describe the complex geometry of the universal Teichmüller space $\mathcal T$, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The quotient $\mathcal S$ of the diffeomorphism group of the circle modulo Möbius …

A Margulis spacetime is a complete flat affine Lorentzian 3-manifold with free fundamental group. Associated to $M$ is a noncompact complete hyperbolic surface $Σ$. We study double extensions of $π_1 (M) \cong π_1 (Σ)$ when $Σ$ is homeomorphic to a projective plane minus two discs. We classify proper actions of this do…

We study subgroups of ${\rm PU}(2,1)$ generated by two non-commuting unipotent maps $A$ and $B$ whose product $AB$ is also unipotent. We call $\mathcal{U}$ the set of conjugacy classes of such groups. We provide a set of coordinates on $\mathcal{U}$ that make it homeomorphic to $\mathbb{R}^2$ . By considering the actio…

The topology of the domain of outer communication for 5-dimensional stationary bi-axisymmetric black holes is classified in terms of disc bundles over the 2-sphere and plumbing constructions. In particular we find an algorithmic bijective correspondence between the plumbing of disc bundles and the rod structure formali…

The main theorem of this paper is a generalisation of well known results about Dehn surgery to the case of attaching handlebodies to a simple 3-manifold. The existence of a finite set of `exceptional' curves on the boundary of the 3-manifold is established. Provided none of these curves is attached to the boundary of a…

The paper explores the topology of polygonal meshes and their properties.

A Margulis spacetime is a complete affine 3-manifold M with nonsolvable fundamental group. Associated to every Margulis spacetime is a noncompact complete hyperbolic surface S. We show that every Margulis spacetime is orientable, even though S may be nonorientable. We classify Margulis spacetimes when S is homeomorphic…

Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.

In this paper, we prove that each injective simplicial map of the arc complex of a compact, connected, orientable surface with nonempty boundary is induced by a homeomorphism of the surface. We deduce, from this result, that the group of automorphisms of the arc complex is naturally isomorphic to the extended mapping c…

The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.

Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.

New knots found with tough, unsliceable discs.

Classifies ancient flows in a disc with boundary.

The paper explores isometric models and Busemann functions for Funk and Hilbert discs.

Let $Γ$ be either the infinite cyclic group $\mathbb{Z}$ or the Baumslag-Solitar group $\mathbb{Z} \ltimes \mathbb{Z}[\frac{1}{2}]$. Let $K$ be a slice knot admitting a slice disc $D$ in the 4-ball whose exterior has fundamental group $Γ$. We classify the $Γ$-homotopy ribbon slice discs for $K$ up to topological ambien…

The rotation angle of a rolling disc is shown to be a geometric phase related to the Hopf fibration.

Study smooth manifolds using disc-presheaves.

This note characterizes monohedral tilings of regular polygons with up to three tiles.

Study horizontal discs in fat distributions, proving their existence.

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.

Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.

The paper classifies homotopy ribbon discs with specific groups.

Study calculates homotopy groups and derivatives for disc diffeomorphisms.

It is well-known that Teichmuller discs that pass through "integer points'' of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of these special Teichmuller discs is mostly unexplored: their number, genus, area, cusps, etc. We prove that in genus two …

Study of rotation angles in a rotating disc model.

The study finds conditions for free boundary Hamiltonian stationary discs in complex 2-space.

Algebraic treatment of connection reduction over a special disc.

We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used …

The paper studies the automorphism groups of specific 3-manifolds and finds upper bounds for their sizes.

Holomorphic discs converge to maximal surfaces under specific flows.

Classifies $4$-manifolds with specific properties and fundamental group.

A subset $E$ of a metric space $X$ is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $\mathbb{R}^n$ for some $n$, sending $E$ to a starlike set. A subset $E\subset X$ is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed s…

The paper finds the Finsler structure of Apollonian weak metric on unit disc.

This paper compares two methods for training neural ODEs in time-series regression and CNFs.

We show that a relatively hyperbolic graph with uniformly hyperbolic peripheral subgraphs is hyperbolic. As an application, we show that the disc graph and the electrified disc graph of a handlebody H of genus g>1 are hyperbolic, and we determine their Gromov boundaries.

In this note we prove that any monohedral tiling of the closed circular unit disc with $k \leq 3$ topological discs as tiles has a $k$-fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain t…

Constructs infinite-dimensional Siegel disc as symplectic and Kaehler quotient.

New phenomena in 4-manifolds show discs with special properties.