New knot graphs show most are not Gromov hyperbolic, with special cases.
problem Characterizing Gromov hyperbolicity in knot graphs.
method Defining knot graphs and proving non-hyperbolicity.
result Most knot graphs are not Gromov hyperbolic, with exceptions.
Gromov boundary described for ray graph action.
problem Understanding the boundary of ray graph.
method Description of Gromov boundary in terms of cliques of long rays.
result Gromov boundary is homeomorphic to a subset of the circle.
The study shows that certain curve graphs are hierarchically hyperbolic but not Gromov hyperbolic.
problem Characterizing the hyperbolicity of curve graphs and their boundaries.
method Using hierarchical hyperbolicity and framed curves, the study examines the properties of curve graphs and their boundaries.
result The curve graphs and their boundaries are hierarchically hyperbolic but not Gromov hyperbolic.
The paper characterizes hyperbolic manifolds and graphs verifying a specific isoperimetric inequality.
problem Understanding the relationship between hyperbolicity and isoperimetric inequalities in manifolds and graphs.
method Characterization of hyperbolic manifolds and graphs with isoperimetric inequality, using Gromov boundary.
result Having a pole is a necessary condition for verifying the isoperimetric inequality, which can be removed.
Upper bounds on topological dimensions of certain graph boundaries.
problem Calculating topological dimensions of specific graph boundaries.
method Linear upper bounds in terms of rank.
result Linear bounds on topological dimensions, dependent on rank.
In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we…
Uniform hyperbolicity proved for nonorientable surface curve graphs.
problem Proving uniform hyperbolicity for nonorientable surface curve graphs.
method Using bicorn curves and arguments from orientable surfaces.
result Graph of nonseparating curves is uniformly hyperbolic.
We show that a relatively hyperbolic graph with uniformly hyperbolic peripheral subgraphs is hyperbolic. As an application, we show that the disc graph and the electrified disc graph of a handlebody H of genus g>1 are hyperbolic, and we determine their Gromov boundaries.
Potential theory extended to Gromov hyperbolic spaces.
problem Extending potential theory to a new class of spaces.
method Unified framework for Gromov hyperbolic metric measure spaces.
result Boundary Harnack inequalities and complete classification of positive harmonic functions.
Defined a new graph type for compact surfaces, proving its connectedness and infinite diameter.
problem Understanding the structure of arc graphs on compact surfaces.
method Defining and analyzing the prescribed arc graph A(Σ,Γ) for compact surfaces Σ with boundary and relations Γ. result The prescribed arc graph A(Σ,Γ) is connected and infinite-diameter, with specific conditions for Gromov hyperbolicity. Boundary rigidity defined for hyperbolic spaces, tied to geometric properties.
problem Understanding boundary rigidity in Gromov hyperbolic spaces.
method Analyzing properties of Gromov hyperbolic spaces and their boundaries.
result Boundary rigidity is equivalent to positive Cheeger isoperimetric constant and non-amenability.
New trick builds hyperbolic manifolds from compact ones, proving some don't virtually fiber.
problem Proving some hyperbolic manifolds don't virtually fiber.
method Hyperbolic reflection group trick, embedding theory, manifold topology.
result Constructed Gromov hyperbolic 7-manifolds that don't virtually fiber over a circle.
We show that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic groups over finite subgroups. Finally, we give a necessary and sufficient condi…
New graph-based complexity measure for hyperbolic 3-manifolds.
problem Defining a complexity measure for hyperbolic 3-manifolds.
method Using Morse functions to graphs and Gromov area.
result Linear relationship between graph adjacency and metric complexity.
Study the geometry of nonseparating curves on surfaces with punctures.
problem Investigate the geometry of nonseparating curves on surfaces with punctures.
method Study finite covers and lifts of nonseparating curves, analyze bicorn curves and laminations.
result Lifts of nonseparating curves span quasiconvex subgraphs in the nonseparating curve graph of covers.
Study shows saddle connection graph's geometry and quasi-isometry properties.
problem Characterize the geometry and quasi-isometry of saddle connection graphs.
method Proved 4-hyperbolicity and uniform quasi-isometry to a tree, used generalised unicorn paths.
result Saddle connection graph is not quasi-isometrically rigid and its boundary is straight foliations.
The grand arc graph's asymptotic dimension is shown to be infinite.
problem Determining the asymptotic dimension of the grand arc graph.
method Using Gromov-hyperbolic and cocompact arc and curve models, the asymptotic dimension is shown to be infinite for a broad class of surfaces.
result The asymptotic dimension of the grand arc graph is infinite.
We prove that the curve graph $\calC^{(1)}(S)$ is Gromov-hyperbolic with a constant of hyperbolicity independent of the surface S. The proof is based on the proof of hyperbolicity of the free splitting complex by Handel and Mosher, as interpreted by Hilion and Horbez.
The aim of this paper is to clarify the relationship between Gromov-hyperbolicity and amenability for planar maps.
The ray graph of a surface with a Cantor set has infinite diameter and is hyperbolic.
problem Understanding the hyperbolicity and quasimorphisms on infinite-type surfaces.
method Action of the mapping class group on the ray graph, explicit construction of quasimorphisms.
result The mapping class group has infinite dimensional second bounded cohomology and vanishing stable commutator length.
Quotients of Gordian and H(2)-Gordian graphs are hyperbolic.
problem Investigate quotients of Gordian and H(2)-Gordian graphs under knot invariants.
method Defined equivalence relations by knot invariants (det, Jones span, tricolorability) and showed quotient graphs are Gromov hyperbolic.
result Quotients of H(2)-Gordian graph of links modulo span of Jones polynomial is isomorphic to complete graph.
The paper studies Lipschitz equivalence of self-similar sets and their augmented trees.
problem Lipschitz equivalence of self-similar sets and their boundaries.
method Introducing simple augmented trees and using combinatorial devices to show Lipschitz equivalence.
result Lipschitz equivalence of self-similar sets and their boundaries.
Characterizes when geodesics in groups are generic.
problem Understanding genericity of geodesics in groups.
method Characterizes Gromov hyperbolicity via contracting elements.
result Genericity of contracting geodesics in groups.
Percolation study in non-hyperbolic groups proves non-uniqueness phase.
problem Percolation in acylindrically hyperbolic groups.
method Analyzing Bernoulli bond percolation on Cayley graphs of groups.
result Non-uniqueness phase in percolation on Cayley graphs of acylindrically hyperbolic groups.
Graphs of multicurves are hierarchically hyperbolic spaces.
problem Understanding the geometric properties of graphs related to surfaces.
method Demonstrating hierarchical hyperbolicity and coarse median properties.
result Graphs of multicurves have a quadratic isoperimetric inequality and are Gromov hyperbolic under certain conditions.
We define analogues of the graphs of free splittings, of cyclic splittings, and of maximally-cyclic splittings of FN for free products of groups, and show their hyperbolicity. Given a countable group G which splits as G=G1∗⋯∗Gk∗F, where F denotes a finitely generated free group, we identify th…
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…
Let S be a surface with genus g and n boundary components and let d(S) = 3g-3+n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions CP(S) prove that the Weil-Petersson metric on Teichmuller space Teich(S) is Gromov-hyperbolic if and only if d(S) …
Study of two actions of mapping class groups on a graph and circle.
problem Understanding dynamics of mapping class groups on graphs and circles.
method Definition and proof of equators, hyperbolic graph, and circle embedding; construction of quasimorphisms.
result Loxodromic elements in the first action have rational rotation numbers in the second action.
The study finds surface subgroups in specific types of groups.
problem Finding surface subgroups in certain groups.
method Analyzing graph pairs and using properties of fundamental groups and limit groups.
result Surface subgroups found in graph pairs and limit groups.
The paper proves drilled bundles over graphs are virtually special cubulable.
problem Proving drilled bundles over graphs are virtually special cubulable.
method Starting with a Gromov-hyperbolic surface bundle, drilling out essential curves, and using relative hyperbolicity and Wise's theorem.
result Proves drilled bundles over graphs are virtually special cubulable.
A Thurston map is a branched covering map from §2 to §2 with a finite postcritical set. We associate a natural Gromov hyperbolic graph $\G=\G(f,\mathcal C)$ with an expanding Thurston map f and a Jordan curve C on §2 containing $\post(f)$. The boundary at infinity of $\G$ with associated visual me…
We show that two uniform lattices of a regular right-angled Fuchsian building are commensurable, provided the chamber is a polygon with at least six edges. We show that in an arbitrary Gromov-hyperbolic regular right-angled building associated to a graph product of finite groups, a uniform lattice is commensurable with…
Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on H_2(G;R) is a finite-sided rational polyhedron.
The paper classifies when certain graph braid groups are 3-manifold groups.
problem Identifying when graph braid groups are 3-manifold groups.
method Analyzing the graph braid groups B3(Θm) for specific graphs Θm. result The paper shows that B3(Θ5) is a 3-manifold group, but B3(Θm) is not quasi-isometric to a 3-manifold group for m≥7. The paper proves Gromov hyperbolicity of certain metrics using isoperimetric inequalities.
problem Investigating Gromov hyperbolicity of specific metrics.
method Using isoperimetric inequalities to characterize Gromov hyperbolicity.
result Characterization of domains where these metrics are Gromov hyperbolic.
Simple Euclidean models outperform hyperbolic graph learning models.
problem The effectiveness of hyperbolic graph learning models is questioned.
method Careful analysis of hyperbolic graph representation learning, identifying and addressing issues with baselines, modeling assumptions, and metric usage.
result Simple Euclidean models often outperform hyperbolic graph learning models, even on hyperbolic datasets.
Surjectivity of Cannon-Thurston map proven for metric graph bundles.
problem Proving surjectivity of Cannon-Thurston map in metric graph bundles.
method Generalized Mj-Sardar's result to include more types of fibers.
result Continuous extension map between boundaries is surjective.
Study the boundary of hyperbolic groups generated by atoroidal automorphisms.
problem Characterize the Gromov boundary of hyperbolic groups generated by atoroidal automorphisms.
method Define directional Whitehead graphs and prove properties of indecomposable trees. Use these to show boundary homeomorphism to Menger curve.
result The boundary of hyperbolic groups generated by atoroidal, fully irreducible automorphisms is homeomorphic to the Menger curve.
A simplicial complex is called negatively curved if all its simplices are isometric to simplices in hyperbolic space, and it satisfies Gromov's Link Condition. We prove that, subject to certain conditions, a compact graph of spaces whose vertex spaces are negatively curved 2-complexes, and whose edge spaces are points …
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…
The study proves Gromov hyperbolicity for certain complex domains.
problem Characterizing Gromov hyperbolicity for complex domains.
method Analyzing domains in C2 with finite d'Angelo type and using automorphisms. result Domains in C2 with finite d'Angelo type are Gromov hyperbolic. Study on non-Gromov hyperbolic tube domains and their geometric properties.
problem Characterizing non-Gromov hyperbolic tube domains with convex bases.
method Provided a criterion for non-Gromov hyperbolicity, studied Hilbert metric, and continuity properties of complex geodesics.
result Similarity of geometry of tube domains and convex domains, connections between metrics.
The paper examines hyperbolicity in bounded strongly minimally convex domains in R^d.
problem Investigating hyperbolicity in bounded strongly minimally convex domains.
method Analyzing the minimal metric and Hilbert metric in convex domains.
result Every bounded strongly minimally convex domain is Gromov hyperbolic.
Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v^v such manifolds of volume at most v, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, alm…
We introduce the co-surface graph CS of a finitely generated free group F and use it to study the geometry of hyperbolic group extensions of F. Among other things, we show that the Gromov boundary of the co-surface graph is equivariantly homeomorphic to the space of free arational $\ma…
Survey solves curvature problems with hyperbolic spaces.
problem Singularities in hypersurface geometry.
method Hyperbolic unfolding correspondence linking hypersurfaces to Gromov hyperbolic spaces.
result Eliminates hypersurface singularities in scalar curvature geometry.
Metric transforms make Euclidean half lines hyperbolic, preserving geodesic properties.
problem Characterizing metric transforms that make Euclidean half lines hyperbolic.
method Characterization of metric transforms φ such that ([0,∞),∣⋅∣φ) is Gromov hyperbolic. result Metric transform rigidity for roughly geodesic Gromov hyperbolic spaces.