arXiv research

Geodesic graphs for special Finsler metrics on spheres are studied.

Optimizing quantum graphs yields geodesic nets on surfaces.

Study geodesics on graphs with random lengths, proving bi-infinite paths exist.

Researchers describe the Gromov boundary of a graph related to surfaces.

Researchers analyze geodesic complexity in robot paths on tree graphs.

We study the geometry of the Thurston metric on Teichmuller space by examining its geodesics and comparing them to Teichmuller geodesics. We show that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to the curve graph is a reparametrized quasi-geodesic. However, we show that the set of short curve…

Study efficient geodesics in curve complex using dot graphs.

In this paper we show that two Lagrangian graphs over the torus in $\mathbb{C}^n$ with large Lagrangian phase can be connected via Lipschitz continuous geodesic with respect to the $L^2$ metric on the space of Lagrangian submanifolds. In particular, the geodesic for Lagrangian graphs over the torus in $\mathbb{C}^n$ ca…

FPP preserves sublinear Morse boundaries in geodesic graphs.

The curve graphs are not locally finite. In this paper, we show that the curve graphs satisfy a property which is equivalent to graphs being uniformly locally finite via Masur--Minsky's subsurface projections. As a direct application of this study, we show that there exist computable bounds for Bowditch's slices on tig…

This paper restricts efficient geodesics to non-separating curves.

We show that for a surface S, the subgraph of the pants graph determined by fixing a collection of curves that cut S into pairs of pants, once-punctured tori, and four-times-punctured spheres is totally geodesic. The main theorem resolves a special case of a conjecture made by Aramayona, Parlier, and Shackleton and has…

The main theorem of this paper classifies the quasi-geodesics in a Coxeter group that are tracked by geodesics. As corollaries, we show that if a Coxeter group acts geometrically on a CAT(0) space X then CAT(0) rays (and lines) are tracked by Cayley graph geodesics, all special subgroups of the Coxeter group are quasi-…

For the pants graph, there is little known about the behaviour of geodesics, as opposed to quasigeodesics. Brock-Masur-Minsky showed that geodesics or geodesic segments connecting endpoints satisfying a bounded combinatorics condition, such as the stable/unstable laminations of a pseudo-Anosov, all have bounded combina…

New definition of naturally reductive Finsler manifolds using geodesic graphs.

We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting of geodesics. Furthermore, the method gives inequalities with a uniform constant…

In this work we study the geodesic flow on nilmanifolds associated to graphs. We are interested in the construction of first integrals to show complete integrability on some compact quotients. Also examples of integrable geodesic flows and of non-integrable ones are shown.

We show that the metric of nonpositively curved graph manifolds is determined by its geodesic flow. More precisely we show that if the geodesic flows of two nonpositively curved graph manifolds are $C^0$ conjugate then the spaces are isometric.

A new discrete formula connects vertex and edge distributions on graphs.

Geodesic nets on flat spheres are studied using Gauss-Bonnet theorem.

We study Riemannian nilmanifolds associated with graphs. We prove that such a nilmanifold is geodesic orbit if and only if it is naturally reductive if and only if its defining graph is the disjoint union of complete graphs and the left-invariant metric is generated by a certain naturally defined inner product.

First-passage percolation affects graph properties like curvature and geodesics.

Study of mean curvature flows on graphs in warped product manifolds, focusing on behavior at infinity.

Asymptotic dimension of planes and graphs is at most three.

Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.

Develops active intervals for geodesics in Teichmüller space.

The study connects geodesic flows on Riemann surfaces to random walks on their dual graphs.

Study geodesics in 3-torus, determining complements' topology.

We give a combinatorial proof, using the hyperbolicity of the curve graphs, of the bounded geodesic image theorem of Masur and Minsky. Recently it has been shown that curve graphs are uniformly hyperbolic, thus a universal bound can be given for the diameter of the geodesic image. We also generalize the theorem for pro…

Geodesics count exponentially between triangulations of surfaces with enough topology.

The study combines graph-minors and metric spaces, answering some questions and conjectures.

We characterize strongly Morse quasi-geodesics in Outer space as quasi-geodesics which project to quasi-geodesics in the free factor graph. We define convex cocompact subgroups of $Out(F_n)$ as subgroups such that an orbit map in the free factor graph is a quasi-isometric embedding, and we characterize such groups via …

Theorem shows generic metrics yield non-degenerate geodesic nets.

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…

The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.

Counting periodic geodesics of bounded length and commutator structure on hyperbolic surfaces.

We propose a novel node embedding of directed graphs to statistical manifolds, which is based on a global minimization of pairwise relative entropy and graph geodesics in a non-linear way. Each node is encoded with a probability density function over a measurable space. Furthermore, we analyze the connection between th…

We find an upper bound for the asymptotic dimension of a hyperbolic metric space with a set of geodesics satisfying a certain boundedness condition studied by Bowditch. The primary example is a collection of tight geodesics on the curve graph of a compact orientable surface. We use this to conclude that a curve graph h…

Random walks on mapping class groups identified with geodesic laminations.

Develops a method to construct entire minimal graphs of odd dimensions.

We prove that a foliation $(M, F)$ of codimension $q$ on a $n$-dimen\-sio\-nal pseudo-Riemannian manifold is pseudo-Riemannian if and only if any geodesic that is orthogonal at one point to a leaf is orthogonal to every leaf it intersects. We show that on the graph $G = G(F)$ of a pseudo-Riemannian foliation there exis…

The paper studies hyperbolic phenomena on closed surfaces using bicorn curves.

Our main theorem asserts that every Farey graph embedded in the 1-skeleton of the pants complex of any finite type surface is totally geodesic.

Analyzes geodesic lengths in sparse networks, deriving a distribution.

New method constructs asymptotic convex hypersurfaces via equidistant hyperplanes.

Study geodesic paths on flat surfaces, comparing length and singularity counts.

New rays on infinite type surfaces help understand their boundaries.

Topology helps estimate chromatic numbers of random graphs on spheres.