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.
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Graphs of multicurves are hyperbolic, relatively hyperbolic, or thick.
problem Characterizing graphs of multicurves based on their geometric properties.
method Proving graphs of multicurves are hyperbolic, relatively hyperbolic, or thick based on subsurface intersections.
result Geometric characterization of graphs of multicurves.
Non-relatively hyperbolic separating curve graph for surfaces.
problem Classifying hyperbolicity of separating curve graphs.
method Proof of non-relatively hyperbolic property.
result Separating curve graph is not relatively hyperbolic for surfaces with genus ≥ 3 and one boundary component.
Algorithm morphs graphs on hyperbolic surfaces.
problem Morphing graphs on hyperbolic surfaces.
method Generalization of Tutte's spring embedding theorem.
result First algorithm for morphing graphs on hyperbolic surfaces.
Hensel-Przytycki-Webb proved that all curve graphs of orientable surfaces are 17-hyperbolic. In this paper, we show that curve graphs of non-orientable surfaces are 17-hyperbolic by applying Hensel-Przytycki-Webb's argument. We also show that arc graphs of non-orientable surfaces are 7-hyperbolic, and arc-curve graphs …
New findings on hyperbolicity of fine curve graphs and their subgraphs.
problem Investigating hyperbolicity of fine curve graphs and their subgraphs.
method Analyzing large subgraphs of fine curve graphs and computing distances in specific cases.
result Large subgraphs of fine curve graphs contain flats of every finite dimension, indicating they are not hyperbolic.
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.
New findings on algebraic structure of hyperbolic graph braid groups.
problem Classifying and understanding the algebraic structure of hyperbolic graph braid groups.
method Analyzing specific graph types (sun and pulsar graphs) and proving theorems about their braid groups.
result 3-strand braid groups of sun graphs are free, while most pulsar graphs contain surface subgroups.
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.
This paper calculates hyperbolicity constants for pants and relative pants graphs.
problem Understanding how hyperbolicity constants change based on the surface.
method Study of hyperbolicity constants for pants and relative pants graphs for specific surfaces.
result Calculates hyperbolicity constants for the five-punctured sphere and twice punctured torus.
A novel hyperbolic graph attention network for non-Euclidean graph data.
problem Non-Euclidean graph data requires specialized models to capture its unique properties.
method Employed gyrovector spaces to transform features and hyperbolic proximity attention mechanism for aggregation. Novel acceleration strategy using logarithmic and exponential mappings.
result Demonstrated superior performance on real-world datasets compared to state-of-the-art methods.
HGCN uses hyperbolic geometry to improve graph node embeddings.
problem Distortion in Euclidean embeddings of real-world graphs.
method Derives GCN operations in hyperbolic space and maps Euclidean features to hyperbolic embeddings.
result HGCN achieves up to 63.1% error reduction in ROC AUC for link prediction.
New hyperbolic graph constructed from projections of free splitting graph.
problem Constructing a new hyperbolic graph from projections of free splitting graph.
method Using submanifold projections and geometric realization of free splitting graph.
result A new hyperbolic graph constructed for .
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.
Hyperbolic GNNs improve graph data learning.
problem Learning from graph-structured data.
method Proposes a novel GNN architecture for Riemannian manifolds.
result Hyperbolic GNNs lead to substantial improvements on benchmark datasets.
Graph conditions ensure matching arc complexes are connected and hyperbolic.
problem Conditions for connectedness and hyperbolicity of matching arc complexes.
method Conditions on finite simplicial graphs guaranteeing connectedness and hyperbolicity of matching arc complexes.
result Conditions on finite simplicial graphs ensure connectedness and hyperbolicity of matching arc complexes.
We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.
Stable cylinders found in hyperbolic groups and curve graphs.
problem Torsionfree hyperbolic groups and curve graphs of surfaces have globally stable cylinders.
method Generalised Sageev's construction to improve fine properties of hyperbolic spaces.
result Proved curve graphs of surfaces admit equivariant quasi-isometric embeddings in finite products of quasitrees.
Study shows disk graphs of certain handlebodies have quadratic asymptotic dimension.
problem Understanding the asymptotic dimension of disk graphs in handlebodies.
method Utilized hyperbolic and relatively hyperbolic graph properties, including electrification.
result Asymptotic dimension of disk graphs in handlebodies of genus ≥2 is at most quadratic.
Proposes HypCSE for enhanced hierarchical clustering.
problem Challenges in existing hierarchical clustering methods.
method Hyperbolic Continuous Structural Entropy (HypCSE) neural networks.
result Superior performance on seven datasets.
The paper extends carrier graphs to free groups in hyperbolic 3-space.
problem Defining carrier graphs for free groups in hyperbolic 3-space.
method Generalized carrier graphs definition and proof of existence and finiteness for discrete, faithful, and geometrically finite representations.
result Existence and finiteness of minimal carrier graphs for specified representations.
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.
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.
Paper discusses hyperbolic structures for Artin-Tits groups.
problem No specific problem stated, focuses on hyperbolic structures.
method Algebraic analogues of previously known structures on Artin braid groups.
result Presented several candidates for hyperbolic structures.
New criteria for relative hyperbolicity in hierarchically hyperbolic spaces.
problem Characterizing relative hyperbolicity in hierarchically hyperbolic spaces.
method New formulation of relative hyperbolicity in terms of hierarchy structures, applied to graphs associated to surfaces.
result The separating curve graph of a surface is relatively hyperbolic when the surface has zero or two punctures.
The study proves conditions for hyperbolic isometries on fine curve graphs of higher genus surfaces.
problem Conditions for hyperbolic isometries on fine curve graphs of higher genus surfaces.
method Proves equivalence of conditions involving isotopic maps, pseudo-Anosov maps, and ergodic rotation sets.
result Ergodic homological rotation sets have nonempty interior for certain isotopic maps.
Paper introduces a hyperbolic approach for community detection on graphs.
problem Detecting communities on graphs efficiently and effectively.
method Combines hyperbolic embeddings with Riemannian K-means or mixture models.
result Demonstrates effectiveness through experiments on real-world social networks.
The fine curve graph is hyperbolic and contains all countable graphs as induced subgraphs.
problem Characterizing the structure and properties of fine curve graphs.
method Analyzing the hyperbolicity and induced subgraph properties of fine curve graphs and their direct limits.
result The finitary curve graph has diameter 2, contains every countable graph as an induced subgraph, and has the homeomorphism group of the surface as its automorphism group.
Study the geometry of graph product extension graphs.
problem Properties of graph products.
method Introduce and study the extension graph of graph products of groups.
result Extension graph is isomorphic to crossing graph of a quasi-median graph and exhibits asymptotic dimension similar to quasi-trees.
New graphs show hierarchical hyperbolic properties, extending previous work.
problem Characterizing hierarchically hyperbolic properties of multiarc and curve graphs.
method Analyzing the geometric intersection number and using PMod(S) action.
result Multiarc and curve graphs are hierarchically hyperbolic.
We show that the graphs of nonseparating curves for oriented finite type surfaces are uniformly hyperbolic. Our proof follows the proof of uniform hyperbolicity of the graphs of curves for closed surfaces due to Przytycki-Sisto, while introducing new arguments using homology to certify that certain curves are nonsepara…
This note addresses some questions that arise in the series of works by Kyoji Saito on the growth functions of graphs. We study "hyperbolike" graphs, which include Cayley graphs of hyperbolic groups. We generalize some well-known results on hyperbolic groups to the hyperbolike setting, including rationality of generati…
Infinite hyperbolic knots yield unusual surgeries.
problem Finding knots with specific surgery results.
method Examined infinite families of hyperbolic knots and their surgeries.
result Discovered knots with surgeries producing graph manifolds with five disjoint, non-parallel incompressible tori.
The study shows acylindrical hyperbolicity for Artin groups not associated with joins or cones.
problem Proving acylindrical hyperbolicity for Artin groups of infinite type not associated with joins or cones.
method Developing and extending the clique-cube complex and action studies of Charney and Morris-Wright.
result Acylindrical hyperbolicity demonstrated for Artin groups of infinite type associated with graphs that are not cones.
Graph products inherit Morse local-to-global property from their components.
problem Generalizing local-to-global property to graph products of infinite groups.
method Generalizing maximization procedure for relatively hierarchically hyperbolic groups and showing stable embeddings.
result Graph products of infinite Morse local-to-global groups have the Morse local-to-global property.
A construction of a spatial graph from a strongly invertible knot was developed by the second author, and a necessary and sufficient condition for the given spatial graph to be hyperbolic was provided as well. The condition is improved in this paper. This enable us to show that certain classes of knots can yield hyperb…
Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.
problem Distribution of shapes of complementary subsurfaces in moduli space.
method Study of shapes of complementary subsurfaces in moduli space as boundary lengths go to infinity.
result Random subsurfaces look like random ribbon graphs.
Sharp bounds for spanning tree entropy in planar lattices.
problem Estimating spanning tree entropy in planar lattice graphs.
method Using hyperbolic geometry and polyhedra volumes.
result Proved bounds are easy to compute and provide excellent estimates.
New research shows hyperbolic embeddings are useful for global consistency tasks in graphs.
problem The usefulness of hyperbolic representations in graph learning tasks.
method Computed hyperbolic embeddings for node classification and link prediction tasks, addressing optimization issues at zero curvature.
result Hyperbolic embeddings are more effective for tasks requiring global consistency, while Euclidean models are superior for other tasks.
MuRP embeds multi-relational graphs in hyperbolic space for better hierarchical representation.
problem Current hyperbolic models struggle with multi-relational knowledge graphs that exhibit multiple hierarchies.
method MuRP embeds multi-relational graph data in the Poincaré ball model of hyperbolic space, learning relation-specific parameters for entity embeddings.
result MuRP embeddings outperform Euclidean counterparts and other methods on link prediction tasks, especially at lower dimensions.
Study shows RAAG automorphisms and outer automorphisms are not relatively hyperbolic.
problem Characterizing automorphism and outer automorphism groups of RAAGs.
method Analyzing groups of RAAGs with at least 3 vertices, categorizing based on graph structure.
result Automorphism and outer automorphism groups of RAAGs are not relatively hyperbolic.
We show that many graphs naturally associated to a connected, compact, orientable surface are hierarchically hyperbolic spaces in the sense of Behrstock, Hagen and Sisto. They also automatically have the coarse median property defined by Bowditch. Consequences for such graphs include a distance formula analogous to Mas…
We prove that the curve graph $\calC^{(1)}(S)$ is Gromov-hyperbolic with a constant of hyperbolicity independent of the surface . The proof is based on the proof of hyperbolicity of the free splitting complex by Handel and Mosher, as interpreted by Hilion and Horbez.
Study connects flow dynamics to 3D geometry via surface intersections.
problem Relating flow dynamics to geometric properties of 3-manifolds.
method Relates pseudo-Anosov flow dynamics to hyperbolic geometry via curve graphs.
result Established a link between flow invariants and geometric features of 3-manifolds.
Corrects a mistake in a theorem about hyperbolic groups.
problem A theorem about graphs of hyperbolic groups was weakened.
method Identified and corrected a mistake in the published paper.
result Proved a weaker result to correct the main theorem.
Researchers prove constant mean curvature graphs in hyperbolic 3-space for specific domains.
problem Existence of hyperbolic Killing graphs with constant mean curvature in exterior domains.
method Existence proof using CMC graphs and Killing vector fields.
result Existence of hyperbolic Killing graphs of constant mean curvature H in exterior domains.
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.
The study proves conjecture for specific Artin groups.
problem Proving conjecture about Artin groups' properties.
method Analyzing Artin groups associated to triangle-free graphs and cones over square-free bipartite graphs.
result Proves conjecture for specific Artin groups.