arXiv research

The study restricts groups in graph of groups structures.

New triangulations of octonionic projective plane found with restricted symmetry groups.

We study the {\it arc and curve} complex $AC(S)$ of an oriented connected surface $S$ of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then the simplicial automorphism group of $AC(S)$ coincides with the natural image of the exten…

Let $Γ$ be a one-ended, torsion-free hyperbolic group and let $G$ be a semisimple Lie group with finite center. Using the canonical JSJ splitting due to Sela, we define amalgam Anosov representations of $Γ$ into $G$ and prove that they form a domain of discontinuity for the action of $\mathrm{Out}(Γ)$. In the appendix,…

Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex in vertex groups, we prove that vertex groups are relatively quasiconvex if and only if all the vertex groups have finite relative height in the f…

Let G and F be finitely generated groups with infinitely many ends and let A and B be graph of groups decompositions of F and G such that all edge groups are finite and all vertex groups have at most one end. We show that G and F are quasi-isometric if and only if every one-ended vertex group of A is quasi-isometric to…

Defines formal vertex laws related to Lie conformal algebras.

We study relationships between the restricted unrolled quantum group $\overline{U}_q^H(\mathfrak{sl}_2)$ at $2r$-th root of unity $q=e^{πi/r}, r \geq 2$, and the singlet vertex operator algebra $\mathcal M(r)$. We use deformable families of modules to efficiently compute $(1, 1)$-tangle invariants colored with projecti…

Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. Fleming and the author introduced some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the constituent 2-component algebraically split li…

Proves a generalized Whitehead cut vertex lemma for tree groups.

634 vertex-transitive and over 10^103 non-vertex-transitive 27-vertex triangulations of octonionic projective plane.

Given a graph in which a few vertices are deemed interesting a priori, the vertex nomination task is to order the remaining vertices into a nomination list such that there is a concentration of interesting vertices at the top of the list. Previous work has yielded several approaches to this problem, with theoretical re…

The paper studies graph products of groups and recovers graph and vertex groups under certain conditions.

Consider a group G and a family $\mathcal{A}$ of subgroups of G. We say that vertex finiteness holds for splittings of G over $\mathcal{A}$ if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal G-trees with edge stabilizers in $\mathcal{A}$. We show vertex finiteness when G…

Connected components of Morse boundaries are studied in graph of groups.

This paper shows semi-equivelar toroidal maps are vertex-transitive covers.

We develop an algorithm of polynomial time complexity to construct the Grushko decomposition of fundamental groups of graphs of free groups with cyclic edge groups. Our methods rely on analysing vertex links of certain CAT(0) square complexes naturally associated with a special class of the above groups. Our main resul…

Finite stature proven for cube complexes with cyclonormal edges.

Let $M$ be a Riemannian manifold. For $p\in M$, the tensor algebra of the negative part of the (complex) affinization of the tangent space of $M$ at $p$ has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over $M$ with a connection. We …

We describe an algorithm for the enumeration of (candidates of) vertex-transitive combinatorial $d$-manifolds. With an implementation of our algorithm, we determine, up to combinatorial equivalence, all combinatorial manifolds with a vertex-transitive automorphism group on $n\leq 13$ vertices. With the exception of act…

Quasi-vertex-transitive maps are the homogeneous maps on the plane with finitely many vertex orbits under the action of their automorphism groups. We show that there exist quasi-vertex-transitive maps of types $[p^3, 3]$ for $p \equiv 1$ (mod $6$), but there doesn't exist vertex-transitive map of such types. In particu…

We prove that the Cayley graph and the coset geometry of the von Dyck group $D(a,b,c)$ are linked by a vertex-to-edge duality.

The study finds the bounds of vertex orbits in maps derived from specific lattices.

We give a technical result that implies a straightforward necessary and sufficient conditions for a graph of groups with virtually cyclic edge groups to be one ended. For arbitrary graphs of groups, we show that if their fundamental group is not one-ended, then we can blow up vertex groups to graphs of groups with simp…

Artin groups have finite stature based on vertex groups.

Generalized Baumslag-Solitar groups are defined as fundamental groups of graphs of groups with infinite cyclic vertex and edge groups. Forester proved (in "On uniqueness of JSJ decompositions of finitely generated groups", Comment. Math. Helv. 78 (2003) pp 740-751) that in most cases the defining graphs are cyclic JSJ …

This paper constructs quandles with abelian inner automorphism groups from graphs, proving their homogeneity.

Abstract studies 3-manifolds and vertex algebras, expanding known connections.

Suppose a finitely generated group $G$ is hyperbolic relative to $\mathcal P$ a set of proper finitely generated subgroups of $G$. Established results in the literature imply that a "visual" metric on $\partial (G,\mathcal P)$ is "linearly connected" if and only if the boundary $\partial (G,\mathcal P)$ has no cut poin…

New triangulations of quaternionic projective plane found with various symmetry groups.

Artin groups of type $D_n$ have special cycles and complexes with interesting properties.

This paper addresses questions of quasi-isometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the Bass-Serre tree of the graph of groups has finite depth. The main example of a finite depth graph of groups is one whose vertex and edge groups are coarse Poinca…

A piecewise flat Finsler metric on a triangulated surface $M$ is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of geodesics on a piecewise flat Finsler surface, especially when it meets a vertex…

We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The ma…

We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more generally, we do not impose any boundedness assumption on the geometry. This is achieved …

Let $Γ$ be a word hyperbolic group with a cyclic JSJ decomposition that has only rigid vertex groups, which are all fundamental groups of closed surface groups. We show that any group $H$ quasi-isometric to $Γ$ is abstractly commensurable with $Γ$.

Graph neural networks suffer from oversmoothing, but adding residual connections helps.

Mathematical construction of vertex algebra representations from integrable G2 structures.

We show how to construct an N=1 superconformal vertex algebra (SCVA) from any Riemannian manifold. When the Riemannian manifold has special holonomy groups, we discuss the extended supersymmetry. When the manifold is complex or Kähler, we also generalize the construction to obtain N=2 SCVA's. We study the BRST cohomolo…

A vertex-transitive map $X$ is a map on a closed surface on which the automorphism group ${\rm Aut}(X)$ acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse…

Two spectral algorithms for community detection in graphs with covariates are compared.

The paper explores new phenomena in boundaries of relatively hyperbolic groups.

In this paper, we classify all of the five-sided three-dimensional hyperbolic polyhedra with one ideal vertex, which have the shape of a triangular prism. We show how to find each such polyhedron in the upper half-space model by considering lines and circles in the plane. Finally, we give matrix generators in $\mathrm{…

Graphically discrete groups have strong rigidity properties.

Assume that Γ_{v_0} is a tree with vertex set Vert(Γ_{v_0})={v_0, v_1,..., v_n}, and with an integral framing (weight) attached to each vertex except v_0. Assume furthermore that the intersection matrix of G=Γ_{v_0}-{v_0} is negative definite. We define a filtration on the chain complex computing the lattice homology o…

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relativ…

We construct examples of finitely generated groups L that have non-trivial actions on $\mathbb{R}$-trees but which cannot act, without fixing a vertex, on any simplicial tree. Moreover, any finitely presented group mapping onto L does have a fixed point-free action on some simplicial tree.

The Four Vertex Theorem, one of the earliest results in global differential geometry, says that a simple closed curve in the plane, other than a circle, must have at least four "vertices", that is, at least four points where the curvature has a local maximum or local minimum. In 1909 Syamadas Mukhopadhyaya proved this …