arXiv research

Study covers of Chamanara surface with large Veech groups.

In this work we deal with coverings and actions of Lie group- groupoids being a sort of the structured Lie groupoids. Firstly, we define an action of a Lie group-groupoid on some Lie group and the smooth coverings of Lie group-groupoids. Later, we show the equivalence of the category of smooth actions of Lie group-grou…

New braid group representations into mapping class groups are derived from disk coverings.

Study of fundamental groups of 3D small covers using Morse theory.

The paper finds new graph covers with exceptionally low degree.

The paper explores simple and relatively simple transformation groups and their universal coverings.

Closed self-covering manifolds with abelian fundamental groups fiber over tori.

This paper concerns the class of contractible open 3-manifolds which are ``locally finite strong end sums'' of eventually end-irreducible Whitehead manifolds. It is shown that whenever a 3-manifold in this class is a covering space of another 3-manifold the group of covering translations must be a free group. It follow…

Study homology groups of mapping and Torelli groups for surfaces with abelian covers.

Study covers of surfaces, showing types and properties.

New method shows any triangle group generating pair is related to special coverings.

The paper characterizes geometrically finite surfaces via geodesic covers.

We study the topology of small covers from their fundamental groups. We find a way to obtain explicit presentations of the fundamental group of a small cover. Then we use these presentations to study the relations between the fundamental groups of a small cover and its facial submanifolds. In particular, we can determi…

This article establishes the algebraic covering theory of quandles. For every connected quandle we explicitly construct a universal covering, which in turn leads us to define the algebraic fundamental group as the automorphism group of the universal covering. We then establish the Galois correspondence between connecte…

The paper proves conditions for self-covering manifolds to be fiber bundles over a circle.

Random branched covers of groups are homotopy equivalent to geometrically small cancellation complexes.

The paper studies liftable mapping class groups of cyclic covers of spheres.

For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the actio…

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 …

In "Rips complexes and covers in the uniform category" \cite{Rips} the authors define, following James \cite{J}, covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given…

The paper proves every closed curve on a bouquet of circles lifts to finite covers.

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

We discuss properties of complex algebraic orbifold groups, their characteristic varieties, and their abelian covers. In particular, we deal with the question of (quasi)-projectivity of orbifold groups. We also prove a structure theorem for the variety of characters of normal-crossing quasi-projective orbifold groups. …

Study shows a central limit theorem for random coverings of manifolds with nilpotent groups.

Study braid group actions on exceptional sequences using branched coverings.

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 …

Constructs a topological cover of real line's multiplicative group.

We prove the homotopy invariance of L^2 torsion for covering spaces, whenever the covering transformation group is either residually finite or amenable. In the case when the covering transformation group is residually finite and when the L^2 cohomology of the covering space vanishes, the homotopy invariance was establi…

The study finds a limit on subgroup complexity in hyperbolic 3-manifold groups.

We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1,1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and uniqueness of the n-fold strongly-cyclic branched coverings of (1,1)-knots, thro…

Theory of $H$-graded manifolds and coverings of supermanifolds.

We study various covering spectra for complete noncompact length spaces with universal covers (including Riemannian manifolds and the pointed Gromov Hausdorff limits of Riemannian manifolds with lower bounds on their Ricci curvature). We relate the covering spectrum to the (marked) shift spectrum of such a space. We de…

Surveying methods to create spaces with non-trivial self covers.

The Birman-Hilden theory is extended to infinite type surfaces and branched covers.

For large genus, precise monodromy groups are calculated for surface covers.

In this paper we study the connections between cyclic presentations of groups and branched cyclic coverings of (1,1)-knots. In particular, we prove that every n-fold strongly-cyclic branched covering of a (1,1)-knot admits a cyclic presentation for the fundamental group encoded by a Heegaard diagram of genus n.

For a connected, locally path connected space $X$, let $H$ be a subgroup of the fundamental group of $X$, $π_1(X,x)$. We show that there exists an open cover $\cal U$ of $X$ such that $H$ contains the Spanier group $π({\U},x)$ if and only if the core of $H$ in $π_1(X,x)$ is open in the quasitopological fundamental grou…

New method finds unbranched covers with non-kernel homology.

We say that a cover of surfaces S -> X has the Birman--Hilden property if the subgroup of the mapping class group of X consisting of mapping classes that have representatives that lift to S embeds in the mapping class group of S modulo the group of deck transformations. We identify one necessary condition and one suffi…

We apply the technique of integrable extensions to the symmetry pseudo-group of the r-th mdKP equation. This gives another look on deriving known coverings and allows us to find new coverings for this equation.

Algorithms compute invariants of 4-manifolds as branched covers.

Uniform waist inequalities proven for manifolds with Kazhdan groups in codimension two.

We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spa…

In this paper, we construct new examples of Veech groups by extending Schmithusen's method for calculating Veech groups of origamis to Veech groups of unramified finite coverings of regular 2n-gons. We calculate the Veech groups of certain Abelian coverings of regular 2n-gons by using an algebraic method.

The paper characterizes coverings over the projective plane with minimal defect.

This paper gives a new proof of a result of Geoghegan and Mihalik which states that whenever a contractible open $n$-manifold $W$ which is not homeomorphic to $\mathbf{R}^n$ is a covering space of an $n$-manifold $M$ and either $n \geq 4$ or $n=3$ and $W$ is irreducible, then the group of covering translations injects …

We consider branched coverings which are simple in the sense that any point of the target has at most one singular preimage. The cobordism classes of $k$-fold simple branched coverings between $n$-manifolds form an abelian group $Cob^1(n,k)$. Moreover, $Cob^1(*,k) = \bigoplus_{n=0}^{\infty} Cob^1(n,k)$ is a module over…

We define an infinite series of translation coverings of Veech's double-n-gon for odd n greater or equal to 5 which share the same Veech group. Additionally we give an infinite series of translation coverings with constant Veech group of a regular n-gon for even n greater or equal to 8. These families give rise to expl…