arXiv research

Quotients of Gordian and H(2)-Gordian graphs are hyperbolic.

The study provides interior curvature estimates for convex graphs satisfying a specific quotient equation.

New insights into ends of quotient spaces and graphs.

Random quotients of mapping class groups have rigid properties.

Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. We obtain periodic isotopy classifications for various families of embedded nets with small quotient graphs. We enumerate the 25 periodic isotopy classes of …

Study homology of periodic cell complexes using quotient spaces and spectral sequences.

This article is about the graph genus of certain well studied graphs in surface theory: the curve, pants and flip graphs. We study both the genus of these graphs and the genus of their quotients by the mapping class group. The full graphs, except for in some low complexity cases, all have infinite genus. The curve grap…

The (torsion) complexity of a finite edge-weighted graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When G is d-periodic (i.e., G has a free action of the rank-d free abelian group by graph automorphisms, with finite quotient) the Mahler measure of its Lap…

Given a metric space $X$ and a function $f: X \to \mathbb{R}$, the Reeb construction gives metric a space $X_f$ together with a quotient map $X \to X_f$. Under suitable conditions $X_f$ becomes a metric graph and can therefore be used as a graph approximation to $X$. The Gromov-Hausdorff distance from $X_f$ to $X$ is b…

The study connects lattices, Garside structures, and weakly modular graphs.

The study examines the rigidity of mapping class groups under large powers of twists.

The homology of Kontsevich's commutative graph complex parameterizes finite type invariants of odd dimensional manifolds. This {\it graph homology} is also the twisted homology of Outer Space modulo its boundary, so gives a nice point of contact between geometric group theory and quantum topology. In this paper we give…

Complexity of signed graphs linked to Alexander polynomials and Lehmer's question.

Introduces intrinsically Lipschitz graphs in metric spaces.

A new filtration of the spaces of tri-/univalent graphs B_m^u that occur in the theory of finite-type invariants of knots and 3-manifolds is introduced. Combining the results of the two preceding articles, the quotients of this filtration are modeled by spaces of graphs with two types of edges and four types of vertice…

To every half-translation surface, we associate a saddle connection graph, which is a subgraph of the arc graph. We prove that every isomorphism between two saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces. We also investigate the automorphism group of the …

Proves representability of complex semigroup systems.

The paper explores conditions for topological rigidity in quotients of the Davis complex.

We study flip-graphs of triangulations on topological surfaces where distance is measured by counting the number of necessary flip operations between two triangulations. We focus on surfaces of positive genus $g$ with a single boundary curve and $n$ marked points on this curve; we consider triangulations up to homeomor…

In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $p\to 1,$ we ident…

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.

New theorem bounds group quotient size to subgroups index.

Graphs with given k vertices generate an (acyclic) simplicial complex. We describe the homology of its quotient complex, formed by all connected graphs, and demonstrate its applications to the topology of braid groups, knot theory, combinatorics, and singularity theory. The multidimensional analogues of this complex ar…

We study a problem of geometric graph theory: We determine the triply periodic graph in Euclidean 3-space which minimizes length among all graphs spanning a fundamental domain of 3-space with the same volume. The minimizer is the so-called srs network with quotient the complete graph on four vertices $K_4$. The network…

The paper extends arithmetic quotient results to right-angled Artin groups.

The ray graph is a Gromov hyperbolic graph on which the mapping class group of the plane minus a Cantor set acts by isometries. We give a description of the Gromov boundary of the ray graph in terms of cliques of long rays on the plane minus a Cantor set. As a consequence, we prove that the Gromov boundary of the ray g…

The study examines flip-graphs of non-orientable surfaces and their diameters.

We explain and generalise a construction due to Gromov to realise geometric small cancellation groups over graphs of groups as fundamental groups of non-positively curved 2-dimensional complexes of groups. We then give conditions so that the hyperbolicity and some finiteness properties of the small cancellation quotien…

We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that overwhelmingly, the knot graphs are not Gromov hyperbolic, with the exception of a part…

The study proves conjecture for specific Artin groups.

Simplified presentation of symplectic fillings of lens spaces.

Study of one-dimensional non-Hausdorff manifolds and their quotient to CW complexes.

This article addresses the two significant aspects of Ozsváth and Szabó's knot Floer cube of resolutions that differentiate it from Khovanov and Rozansky's HOMFLY-PT chain complex: (1) the use of twisted coefficients and (2) the appearance of a mysterious non-local ideal. Our goal is to facilitate progress on Rasmussen…

For a smoothing Y of a 2-dimensional cyclic quotient singularity X, we construct a simple handle decomposition of Y by using a particular birational map from Y to the projective plane. The manifold Y is built up from the product of an annulus with a disk by attaching 2-handles in a manner which can be described by mean…

Let $n$ be a nonnegative integer, we use ribbon $n-$graph diagrams and the Yamada polynomial skein relations to construct an algebra ${\mathcal Y}_n$ which is shown to be closely related to the Temerley-Lieb Algebra. We prove that the algebra ${\mathcal Y}_2$ is isomorphic to some quotient of a three variables polynomi…

Study lens spaces' definite fillings, classifying those with specific inequalities.

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

Let $W$ be a right-angled Coxeter group corresponding to a finite non-discrete graph $\mathcal{G}$ with at least $3$ vertices. Our main theorem says that $\mathcal{G}^c$ is connected if and only if for any infinite index quasiconvex subgroup $H$ of $W$ and any finite subset $\{ γ_1, \ldots , γ_n \} \subset W \setminus …

We study the set of critical exponents of discrete groups acting on regular trees. We prove that for every real number $δ$ between $0$ and $\frac{1}{2}\log q$, there is a discrete subgroup $Γ$ acting without inversion on a $(q+1)$-regular tree whose critical exponent is equal to $δ$. Explicit construction of edge-index…

We show that the Gromov boundary of the free factor graph for the free group Fn with n>2 generators is the space of equivalence classes of minimal very small indecomposable projective Fn-trees without point stabilizer containing a free factor equipped with a quotient topology. Here two such trees are equivalent if the …

We define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. The n-th subgraph of this family is generated by all knots whose fundamental groups surject to a symmetric group with parameter at least n, where all meridians are mapped to transpositions. Incidentally, we verify the Meridio…

Answering a question asked by Agol and Wise, we show that a desired stronger form of Wise's malnormal special quotient theorem does not hold. The counterexamples are generalizations of triangle groups, built using the Ramanujan graphs constructed by Lubotzky--Phillips--Sarnak.

New stability conditions identified from quadratic differentials on surfaces.

The study connects Kleinian group divergence to random walk recurrence.

Study confined subgroups in groups with contracting elements, showing their growth rate is strictly greater than half of the ambient growth rate.

Explicit presentations found for asymptotically rigid mapping class groups.

Dani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra $\mathfrak{n}_G$ from a simple directed graph $G$ in 2005. There is a natural inner product on $\mathfrak{n}_G$ arising from the construction. We study geometric properties of the associated simply connected 2-step nilpotent Lie group …

The moduli space of Riemann surfaces with at least two punctures can be decomposed into a cell complex by using a particular family of ribbon graphs called Nakamura graphs. We distinguish the moduli space with all punctures labelled from that with a single labelled puncture. In both cases, we describe a cell decomposit…