arXiv research

New proof of graph covering theorem for graphs with fins.

Generalizes Leighton's theorem to cube complexes.

Extends canonical measures to metric graphs and proves a generalized Kazhdan's theorem.

Study the geometry of nonseparating curves on surfaces with punctures.

In this paper, we prove uniform lower bounds on the volume growth of balls in the universal covers of Riemannian surfaces and graphs. More precisely, there exists a constant $δ>0$ such that if $(M,hyp)$ is a closed hyperbolic surface and $h$ another metric on $M$ with $\area(M,h)\leq δ\area(M,hyp)$ then for every radiu…

To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. In this article we prove a sufficient numerical condition on the splice diagram for a graph manifold to be a singularity link. We also show that if two manifolds have the same splice diagram, then their unive…

Generalized Thurston's characterization for branched coverings of the 2-sphere.

Automorphisms of contact graphs match those of ${ m CAT(0)}$ cube complexes under weak conditions.

The paper describes relations between Liouville type theorems for solutions of a periodic elliptic equation (or a system) on an abelian cover of a compact Riemannian manifold and the structure of the dispersion relation for this equation at the edges of the spectrum. Here one says that the Liouville theorem holds if th…

Determinants of theta curves and symmetric graphs are studied.

This is a recreational paper showing that certain linked graphs cannot be separated. The proofs employ elementary covering space theory, an appeal to a theorem of Scharlemann (concerning the band sums of two unknots), and a Jones polynomial calculation.

We show that simple coverings of B^4 branched over ribbon surfaces up to certain local ribbon moves bijectively represent orientable 4-dimensional 2-handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over…

Lectures on symplectic and Poisson geometry, quantization, and quantum field theory.

A {\em solvable} cover of a graph is a regular cover whose covering transformation group is solvable. In this paper, we show that a solvable cover of a graph can be decomposed into layers of abelian covers, and also, a lift of a given automorphism of the base graph of a solvable cover can be decomposed into layers of l…

The paper studies orbifold splice quotients and log covers of surface pairs.

Extends Thurston's combinatorial characterization to all branched coverings of the 2-sphere.

New invariant shows 4D graph-manifolds' covers match orthogonal types.

We define metric bundles/metric graph bundles which provide a purely topological/coarse-geometric generalization of the notion of trees of metric spaces a la Bestvina-Feighn in the special case that the inclusions of the edge spaces into the vertex spaces are uniform coarsely surjective quasi-isometries. We prove the e…

The paper finds new graph covers with exceptionally low degree.

We show that for any n > 3 there exists an equivalence functor from the category of n-fold connected simple coverings of B^3 x [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, to the category Chb^{3+1} of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a c…

In this note, we show that the solution to the Dirichlet problem for the minimal surface system in any codimension is unique in the space of distance-decreasing maps. This follows as a corollary of the following stability theorem: if a minimal submanifold $Σ$ is the graph of a (strictly) distance-decreasing map, then $…

We prove that the universal cover of any graph manifold quasi-isometrically embeds into a product of three trees. In particular we show that the Assouad-Nagata dimension of the universal cover of any closed graph manifold is 3, proving a conjecture of Smirnov.

The study extends Kazhdan's theorem to infinite Galois covers of Riemann surfaces.

Graph manifolds' simplicial volume approximated by covering volumes.

We prove the "End Curve Theorem," which states that a normal surface singularity $(X,o)$ with rational homology sphere link $Σ$ is a splice-quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An "end-curve function" is an analytic function $(X,o)\to (\C,0)$ whose ze…

Extends Gromov's theorem with amenable covers.

Paper extends theorem on covering spaces and Jordan curves.

The study connects projective codes to the distribution of zeros of odd maps.

Enhances understanding of stability conditions on surfaces.

Optimizes cover parameter in Mapper algorithm for better visualization.

New graph invariant measures embeddability in 3D.

Let N be a regular branched cover of a homology 3-sphere M with deck group G isomorphic to Z_2^d and branch set a trivalent graph Gamma; such a cover is determined by a coloring of the edges of Gamma with elements of G. For each index-2 subgroup H of G, M_H = N/H is a double branched cover of M. Sakuma has proved that …

New graphs found in CAT(0) group boundaries.

New signatures for knotted graphs linked to classical knot signatures.

Researchers calculate the volume of Seifert representations for graph manifolds and their covers.

Algorithm calculates genus of embedded graphs on surfaces.

The Rips complex at scale r is homotopy equivalent to the nerve of a cover of diameter r.

Determines the type of 3-manifolds from their covers.

The paper proves a theorem about torsion and Morse functions for covering spaces.

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

It was pointed out to us that the proof of a crucial lemma (Lemma 5.3) in the paper is incorrect. Thus the approximation theorem (Theorem 0.1) for L^2 torsion of an amenable covering of a finite simplicial complex remains unproved. However, results and proofs of the first four sections (in particular, the approximation…

This paper extends the Good Covering Theorem and Jordan Curve Theorem for proximal Alexandrov spaces.

Since their introduction by Thurston, geodesic laminations on hyperbolic surfaces occur in many contexts. In this paper, we propose a generalization of geodesic laminations on locally CAT(0), complete, geodesic metric spaces, whose boundary at infinity of the universal cover is endowed with a invariant total cyclic ord…

To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. It was shown earlier that the splice diagram determines the universal abelian cover of the manifold. We will in this article turn the proof of this in to an algorithm to explicitly construct the universal abe…

Support theorem proved for X-ray transform on certain non-compact manifolds.

It was shown in my earlier article that the splice diagram of a rational homology sphere graph manifold determines the manifolds universal abelian cover. In this article we use the proof of this to give a condition on the splice diagram to determine when the universal abelian cover itself is a rational homology sphere.

Random covers of hyperbolic surfaces follow a specific probability measure.

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.