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.
Parabolic mapping class acts on curve graphs of infinite type surfaces.
problem Understanding parabolic isometries on curve graphs of infinite type surfaces.
method Fine curve graph tools to prove existence of parabolic isometries.
result Existence of parabolic isometries on graphs of curves of infinite type surfaces.
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.
Automorphisms of fine 1-curve graph linked to surface homeomorphisms.
problem Understanding automorphisms of fine 1-curve graphs.
method Isomorphic mapping to surface homeomorphisms.
result Automorphism group is isomorphic to homeomorphism group of a surface.
Automorphisms of fine curve graphs match surface homeomorphisms for planar surfaces.
problem Understanding automorphisms of fine curve graphs on surfaces.
method Analyzing vertices and edges of fine curve graphs to match with surface homeomorphisms.
result Automorphism group of fine curve graphs is naturally isomorphic to the homeomorphism group of boundaryless planar surfaces with at least 7 punctures.
Study automorphisms of smooth curve graphs on surfaces.
problem Understanding automorphisms of fine curve graphs.
method Examined automorphisms of continuously differentiable curves on surfaces.
result Automorphisms on surfaces of genus ≥ 2 are induced by homeomorphisms.
Automorphisms of fine curve graph match surface homeomorphisms.
problem Understanding automorphisms of curve graphs for surfaces.
method Building on previous work, proving isomorphism to surface homeomorphisms.
result The group of automorphisms of the fine curve graph is isomorphic to the extended mapping class group of the surface.
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.
Automorphism group of nonorientable surface curve graph matches surface homeomorphisms.
problem Identifying automorphisms of nonorientable surface curve graphs.
method Using Bowden, Hensel, and Webb's fine curve graph and Long, Margalit, Pham, Verberne, and Yao's proof as a foundation.
result Automorphism group of nonorientable surface curve graph is isomorphic to the surface's homeomorphism group.
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.
Study shows surfaces without certain curves have infinite orbit graph.
problem Characterizing surfaces with specific curve properties.
method Utilized tools from mapping class group geometry.
result Infinite-invariance index 1 surfaces lack good curve graphs.
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 …
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…
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.
Determinants of theta curves and symmetric graphs are studied.
problem Understanding the determinants of theta curves and symmetric graphs.
method Combinatorial approach using Kirchhoff's Matrix Tree Theorem and spanning tree enumeration.
result The determinant of a simple theta curve is the product of the determinants of its constituent knots.
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 study arc graphs and curve graphs for surfaces of infinite topological type. First, we define an arc graph relative to a finite number of (isolated) punctures and prove that it is a connected, uniformly hyperbolic graph of infinite diameter; this extends a recent result of J. Bavard to a large class of punctured sur…
We study the chromatic number of the curve graph of a surface. We show that the chromatic number grows like k log k for the graph of separating curves on a surface of Euler characteristic -k. We also show that the graph of curves that represent a fixed non-zero homology class is uniquely t-colorable, where t denotes it…
The curve graph and related graphs are hyperbolic and have quasi-tree fibers.
problem Understanding the structure of the curve graph and related graphs.
method Analyzing a sequence of graphs with Lipschitz maps and proving hyperbolicity and quasi-tree properties.
result The graphs in the sequence are hyperbolic and have quasi-tree fibers, leading to bounds on asymptotic dimension and acylindrical actions.
We investigate the geometry of the graphs of nonseparating curves for surfaces of finite positive genus with potentially infinitely many punctures. This graph has infinite diameter and is known to be Gromov hyperbolic by work of the author. We study finite covers between such surfaces and show that lifts of nonseparati…
New combinatorial type helps distinguish plane curve topologies.
problem Distinguishing the topology of plane curves.
method Introducing G-combinatorial type using modified plumbing graphs.
result Invariant of G-combinatorial type under certain homeomorphisms.
The random graph is an infinite graph with the universal property that any embedding of G−v extends to an embedding of G, for any finite graph. In this paper we show that this graph embeds in the curve graph of a surface Σ if and only if Σ has infinite genus, showing that the curve system on an infinite genus s…
This study exhausts curve graphs of low-genus surfaces.
problem Exhausting curve graphs of low-genus surfaces.
method Constructing finite subgraphs and using rigid expansions.
result Graph morphisms and endomorphisms are automorphisms and induced by homeomorphisms.
Study of graphs interpolating curve and pants graphs, providing formulae and geometry classifications.
problem Understanding the large-scale geometry of graphs connecting curve and pants graphs.
method Developed explicit formulae for quasi-flat ranks and classified geometries using twist-free graphs of multicurves.
result Explicit formulae for quasi-flat ranks and classification of geometries into hyperbolic, relatively hyperbolic, and thick cases.
The paper studies hyperbolic phenomena on closed surfaces using bicorn curves.
problem Understanding hyperbolic phenomena on curve graphs of closed surfaces.
method Using the theory of bicorn curves to analyze the curve graphs of closed surfaces.
result Proves that the curve graph of any closed surface is 15-hyperbolic with one exception.
The study connects spheres in specific surface curve graphs, proving connectivity and classifying components.
problem Proving connectivity and classifying components of spheres in curve graphs of low and medium complexity surfaces.
method Analyzing specific surfaces Σ2,0,Σ1,3,Σ0,6 and Σ0,5,Σ1,2, proving connectivity and classifying components. result Spheres of any radius are connected in Σ2,0,Σ1,3,Σ0,6, and the union of two consecutive spheres is connected in Σ0,5 and Σ1,2. The study examines translation lengths of pseudo-Anosov maps on curve graphs.
problem Estimating translation lengths of pseudo-Anosov maps on curve graphs.
method Analyzing geodesic axes and powers of Dehn twists.
result Determining minimal translation lengths and optimizing map ratios.
New invariant links graph structure to tropical curve properties.
problem Understanding graph and curve minor structures.
method Defined Ceresa-Zharkov class for graphs, related to tropical curves.
result Ceresa-Zharkov class is zero for hyperelliptic graphs.
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.
In this paper, we show that a nontrivial compact graph manifold is nonpositively curved if and only if its fundamental group virtually embeds into a right-angled Artin group. As a consequence, nonpositively curved graph manifolds have linear fundamental groups.
We show that the metric of nonpositively curved graph manifolds is determined by its geodesic flow. More precisely we show that if the geodesic flows of two nonpositively curved graph manifolds are C0 conjugate then the spaces are isometric.
The paper studies the connectedness of a graph's boundary for surfaces.
problem Understanding the topology of the Gromov boundary of fine curve graphs for surfaces.
method Proved a bounded geodesic image theorem, used to show linear connectivity of the Gromov boundary.
result The Gromov boundary of fine curve graphs for surfaces is linearly connected.
Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of such curves admitting representatives that intersect at most k times. We prove that…
We prove that there is an algorithm to determine if a given finite graph is an induced subgraph of a given curve graph.
The splitting number is effective to distinguish the embedded topology of plane curves, and it is not determined by the fundamental group of the complement of the plane curve. In this paper, we give a generalization of the splitting number, called the splitting graph. By using the splitting graph, we classify the embed…
We consider several natural sets of curves associated to a given Teichmüller disc, such as the systole set or cylinder set, and study their coarse geometry inside the curve graph. We prove that these sets are quasiconvex and agree up to uniformly bounded Hausdorff distance. Furthermore, we describe two operations on cu…
The curve graph's model theory reveals its central role in surface study.
problem Why is the curve graph central in surface and mapping class group studies?
method Developed a bridge between model theory, topology, and group theory; bi-interpreted curve graph with mapping class group.
result Proved the curve graph's first-order theory is ω-stable and has quantifier elimination. Spheres in curve graphs are connected, proving Gromov boundary linearity.
problem Understanding connectivity in curve graphs and their boundaries.
method Defining spheres and analyzing their connectivity for different complexities.
result Spheres in high complexity curve graphs are always connected, with weaker results for low complexity.
We estimate the distance in the curve graph of a surface S of finite type using Teichmueller geodesics and assuming to be able to detect curves of distance at least three.
We prove that the separating curve graph of a connected, compact, orientable surface with genus at least 3 and a single boundary component is not relatively hyperbolic. This completes the classification of when the separating curve graph is hyperbolic and relatively hyperbolic initiated by previous works of the authors…
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.
We find an upper bound for the asymptotic dimension of a hyperbolic metric space with a set of geodesics satisfying a certain boundedness condition studied by Bowditch. The primary example is a collection of tight geodesics on the curve graph of a compact orientable surface. We use this to conclude that a curve graph h…
The paper studies curve shortening flows on non-convex surfaces.
problem Behavior of curve shortening flows on non-convex surfaces.
method Defined a graph property and proved its preservation under curve shortening flow.
result The curve becomes a graph after a finite time under the curve shortening flow.
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…
Square inscribed in a curve made of two graph functions.
problem Finding inscribed squares in curves formed by graph functions.
method Analysis of spectral invariants of Jordan Floer homology under curve perturbations.
result Existence of inscribed squares in curves with specific Lipschitz constants.
Connected graph for twice-punctured torus curves.
problem Structure of tri-pants graph on twice-punctured torus.
method Examined relationship with Farey complex to prove connectivity and infinite diameter.
result Tri-pants graph is connected and has infinite diameter.
For an orientable surface S of finite topological type with genus g≥3, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph of S. The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set, and in fact a …
Study the Gromov boundary of fine curve graph for surface homeomorphisms.
problem Understanding the boundary of fine curve graph for surface homeomorphisms.
method Examined the Gromov boundary and local topology near specific foliations and laminations.
result Found elements with positive stable commutator length and proved a Tits alternative.