arXiv research

In this paper, we investigate a relation between finite graphs, simplicial flag complexes and right-angled Coxeter groups, and we provide a class of reconstructible finite graphs. We show that if $Γ$ is a finite graph which is the 1-skeleton of some simplicial flag complex $L$ which is a homology manifold of dimension …

Graph conditions ensure matching arc complexes are connected and hyperbolic.

Automorphisms and subdivisions of Helly graphs are studied, leading to explicit models and rational translation lengths.

Graph manifolds are manifolds that decompose along tori into pieces with a tame $S^1$-structure. In this paper, we prove that the simplicial volume of graph manifolds (which is known to be zero) can be approximated by integral simplicial volumes of their finite coverings. This gives a uniform proof of the vanishing of …

The flip graph and arc complex of a surface are shown to have finite rigidity.

For a finite simplicial graph $Γ$, let $A(Γ)$ denote the right-angled Artin group on $Γ$. Recently Kim and Koberda introduced the extension graph $Γ^e$ for $Γ$, and established the Extension Graph Theorem: for finite simplicial graphs $Γ_1$ and $Γ_2$ if $Γ_1$ embeds into $Γ_2^e$ as an induced subgraph then $A(Γ_1)$ emb…

New simplicial complex for infinite-type surfaces shows graph properties.

The paper provides conditions for realizing graphs and polytopes with specified edge lengths.

Study on finiteness property of right-angled Artin groups actions on extension graphs.

It was proven by González-Meneses, Manchón and Silvero that the extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex obtained from a bipartite circle graph constructed from the diagram. In this paper we conjecture that this simplicial complex is al…

We prove a strong form of finite rigidity for pants graphs of spheres. Specifically, for any $n\geq4$, we construct a finite subgraph $X_n$ of the pants graph $P(S_{0,n})$ of the n-punctured sphere $S_{0,n}$ with the following property. Any simplicial embedding of $X_n$ into any pants graph $P(S_{0,m})$ of a punctured …

New Markov chains defined on simplicial complexes for understanding their topology.

We prove that every injective simplicial map $\mathcal{F}(S) \to \mathcal{F}(S')$ between flip graphs is induced by a subsurface inclusion $S\to S'$, except in finitely many cases. This extends a result of Korkmaz--Papadopoulos which asserts that every automorphism of the flip graph of a surface without boundary is ind…

We study the ideal triangulation graph $T(S)$ of a punctured surface $S$ of finite type. We show that if $S$ is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of $S$ into the simplicial automorphism group of $T(S)$ is an isomorphism…

New homology theory for graphs detects subdivisions and homology manifolds.

Let $S_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We identify a finite rigid subgraph $X_{g,n}$ of the pants graph $\mathcal P (S_{g,n})$, that is, a subgraph with the property that any simplicial embedding of $X_{g,n}$ into any pants graph $\mathcal P (S_{g',n'})$ is induced by an embedding $S_{g…

A notion of up and down Grover walks on simplicial complexes are proposed and their properties are investigated. These are abstract Szegedy walks, which is a special kind of unitary operators on a Hilbert space. The operators introduced in the present paper are usual Grover walks on graphs defined by using combinatoria…

Extends graph degree theorem to simplicial closure of Auter space.

Let $\mathrm{Out}(F_n)$ be the outer automorphism group of the free group $F_n$. It acts properly on the outer space $X_n$ of marked metric graphs, which is a finite-dimensional infinite simplicial complex with some simplicial faces missing. In this paper, we construct complete geodesic metrics and complete piecewise s…

Alexander's conjecture extended to infinite simplicial complexes.

Mixes higher-order simplicial complexes for data augmentation.

This note is about the geometry of the pants graph P(S), a natural simplicial graph associated to a finite type topological surface S where vertices represents pants decompositions. The main result in this note ascserts that for a multicurve Q whose complement is a number of subsurfaces of complexity at most 1. We prov…

Solved Cheeger inequalities for simplicial complexes, combining topological and graph theoretic methods.

BScNets expands graph learning to higher-order interactions.

We study some graphs associated to a surface, called k-multicurve graphs, which interpolate between the curve complex and the pants graph. Our main result is that, under certain conditions, simplicial embeddings between multicurve graphs are induced by $π_1$-injective embeddings of the corresponding surfaces. We also p…

In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These {\it multiarc graphs} naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and topology. We show a number of rigidity results, namely showing that, under certa…

The paper studies connectivity properties of Morse complexes as simplicial complexes grow.

We prove that, except in some low-complexity cases, every locally injective simplicial map between pants graphs is induced by a $π_1$-injective embedding between the corresponding surfaces.

The study examines conditions for minimal volume entropy of simplicial complexes.

Minimal volume entropy vanishes or is positive under certain fiber growth conditions.

We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse of finite spaces induces a simplicial …

This paper classifies planar-Rips complexes and their unit disk graphs up to homotopy.

Groups of homotopy equivalences of graphs help realize compact subgroups.

Simplicial neural networks extend graph neural networks to handle higher-order interactions.

The study shows simplicial volume finiteness for certain manifolds with amenable fundamental 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…

Study simplicial volume of manifolds from reflection group trick.

Finite simplicial complexes dominate certain manifolds with a bounded number of simplices.

Let $M$ be the interior of a connected, oriented, compact manifold $V$ of dimension at least 2. If each path component of $\partial V$ has amenable fundamental group, then we prove that the simplicial volume of $M$ is equal to the relative simplicial volume of $V$ and also to the geometric (Lipschitz) simplicial volume…

The pants graph of a free group is constructed and studied.

This paper extends results of Hatcher and Vogtmann's work "Cerf Theory for Graphs" to ribbon graphs. Given an orientable, punctured and basepointed surface Sigma, we prove that the space of ribbon graphs that can be drawn in Sigma is filtered by simplicial complexes. The k-th simplicial complex is (k-1)-dimensional, (k…

TopoNTK kernel captures higher-order interactions in simplicial complexes.

Involutive Hopf monoids yield surface invariants.

Rust library solves complex equations on abstract simplicial complexes.

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively c…

The study explores discrete versions of Riemannian geometry structures on manifolds.

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for Q-rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the…

Sheaves on graphs link to noncommutative geometry.