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.
Connectivity proven in large rank Gromov boundary of free factor complex.
problem Connectivity of Gromov boundary in large rank free factor complex.
method Analyzing Gromov boundary and free factor complex properties.
result Gromov boundary is path connected and locally path connected in large rank.
Researchers describe the Gromov boundary of a graph related to surfaces.
problem Understanding the Gromov boundary of a graph associated with surfaces.
method Described a dense subset of the Gromov boundary as geodesic laminations, proving the graph satisfies a bounded geodesic image theorem.
result The boundary is not compact.
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.
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.
It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov boundaries. In this paper we introduce the cl…
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.
Proves equivalence of two types of boundaries in metric spaces.
problem Proving equivalence of two types of boundaries in metric spaces.
method Analyzes and compares contracting and κ-Morse boundaries. result Proves equivalence of 1-Morse boundary and contracting boundary as topological spaces.
Compactness theorem for manifolds with boundary proved.
problem Proving convergence of manifolds with boundary.
method Reduction to Hamilton's compactness theorem for manifolds without boundary.
result Cheeger-Gromov convergence for a subsequence of manifolds with bounded geometry.
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. 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.
New index theory proves Gromov's dihedral conjectures.
problem Comparisons and rigidity of scalar curvatures, mean curvatures, and dihedral angles.
method Developed a new index theory for manifolds with polyhedral boundary.
result Proved Gromov's dihedral extremality and rigidity conjectures.
The study describes boundaries of amalgamated products of hyperbolic groups.
problem Understanding the boundaries of amalgamated products of hyperbolic groups.
method Analyzing the Gromov boundary of free products of hyperbolic groups.
result The Gromov boundary of the free product of several hyperbolic groups amalgamated wrt. a common subgroup.
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…
Extends K-cowaist inequality to manifolds with boundary.
problem Applying K-cowaist inequality to manifolds with boundary.
method Extended K-cowaist inequality for generalized Dirac operators on manifolds with boundary.
result New applications of K-cowaist inequality to manifolds with boundary.
The paper examines conditions for the Kobayashi metric to be Gromov hyperbolic on complex convex sets.
problem Conditions for the Kobayashi metric to be Gromov hyperbolic on complex convex sets.
method Develops necessary and sufficient conditions for Gromov hyperbolicity, extends results to C1-smooth boundaries, and provides counterexamples. result Gromov hyperbolicity of Kobayashi metric on C-convex sets with C1-smooth boundary, and counterexamples showing boundary regularity is necessary. 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.
Study defines a new boundary for CAT(0) groups, invariant under quasi-isometries.
problem Defining boundaries for CAT(0) groups when visual boundaries are not well-defined.
method Introducing a sublinear function κ to define κ-Morse boundaries, showing invariance under quasi-isometries.
result κ-Morse boundaries are invariant and metrizable for CAT(0) groups.
We observe after Bayle and Rosales that the Levy-Gromov isoperimetric inequality generalizes to convex manifolds with boundary.
Study transforms singular area minimizing hypersurfaces into hyperbolic spaces.
problem Understanding singular area minimizing hypersurfaces.
method Conformal unfolding to Gromov hyperbolic spaces.
result Recover singular set as Gromov and Martin boundaries.
The study connects hyperbolicity and isoperimetric inequality in manifolds and graphs.
problem Understanding the relationship between hyperbolicity and isoperimetric inequality.
method Characterization of hyperbolic manifolds and graphs with isoperimetric inequality.
result Characterization of trees with isoperimetric inequality and solvability of Dirichlet problem at infinity.
Hyperbolic groups act on spheres, and nearby actions are semi-conjugate.
problem Stability of group actions on spheres.
method Topological stability in dynamical sense.
result Nearby actions are semi-conjugate to the standard boundary action.
The goal of this work is to establish a proof of the Gromov convergence in Hoelder spaces for curves with a totally real boundary condition following the original geometric idea of Gromov. We use a local reflection principle in neighbourhoods of the totally real submanifold as developed by Ivashkovich and Shevchishin a…
A new boundary for geodesic spaces defined and studied.
problem Understanding boundaries of geodesic spaces.
method Definition and study of quasi-geometric boundary ∂QGX. result The quasi-geometric boundary ∂QGX is compact and invariant under quasi-isometric equivalences. Characterizes Coxeter groups with specific boundary shapes.
problem Identifying Coxeter groups with Sierpiński or Menger curve boundaries.
method Combining results from the literature on Gromov boundaries and Coxeter groups.
result Complete characterizations of hyperbolic Coxeter groups with Sierpiński or Menger curve boundaries.
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.
This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The main result e…
This paper extends boundary embedding results to coarsely convex spaces.
problem Generalizing boundary embedding results to coarsely convex spaces.
method Generalizing Dydak and Virk's work on Gromov hyperbolic spaces to coarsely convex spaces.
result Maps between coarsely convex spaces induce continuous maps between their boundaries.
Study G-H limits of surfaces with boundary, focusing on same Euler characteristic.
problem Investigate Gromov-Hausdorff limits of compact surfaces with boundary.
method Focus on surfaces with same Euler characteristic, build on previous work on closed surfaces.
result Complete description and topological properties of limit spaces.
Study limits of curved spaces with boundaries.
problem Understanding geometric structures of curved spaces with boundary constraints.
method Gromov-Hausdorff convergence and collapsing analysis of compact Riemannian manifolds with boundary.
result Describe local geometric structure of limit spaces and establish stability results.
Study Gromov hyperbolic domains in Minkowski space, proving equivalence to boundary properties.
problem Investigate Gromov hyperbolic domains in Minkowski space.
method Explicit comparisons between metrics, dynamical arguments, and quasi-hyperbolic metric.
result Gromov hyperbolicity of convex, future complete domains is equivalent to stable acausality of the boundary.
Trees emerge from group boundaries in hyperbolic geometry.
problem Understanding boundaries of hyperbolic groups and their maps.
method Analyzing short exact sequences of hyperbolic groups and their boundaries.
result The quotient of H boundary by ending laminations is a dendrite.
Compact sequences of manifolds with boundary converge smoothly under conformal transformations.
problem Compactness of sequences of manifolds with boundary under conformal transformations.
method Use of stable elliptic estimates and the 'flatzoomer' method.
result Smooth Cheeger-Gromov convergence of sequences of manifolds with boundary under conformal transformations.
In this note we discuss the behavior of the Gromov boundaries and limit sets for the surface subgroups of the mapping class group with accidental parabolics constructed by the author and A. Reid in earlier work. Specifically, we show that generically there are no Cannon--Thurston maps from the Gromov boundary to Thurst…
Study extends biholomorphisms between convex domains in complex space without boundary constraints.
problem Extending biholomorphisms between convex domains without boundary regularity.
method Combining coarse geometry techniques with dynamical properties of maps in Gromov hyperbolic spaces.
result Proves extensions for biholomorphisms and quasi-isometries between convex domains.
We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on trees. This implies that their Gromov boundaries, defined at density less than 1/2, are Menger curves.
Topology of non-orientable spaces without boundary is studied.
problem Topology of non-collapsed RCD spaces without boundary.
method Studied the stability of non-orientability and topology under Gromov-Hausdorff convergence.
result Non-orientable spaces without boundary have a stable ramified double cover.
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. We show that trees of manifolds, the topological spaces introduced by Jakobsche, appear as boundaries at infinity of various spaces and groups. In particular, they appear as Gromov boundaries of some hyperbolic groups, of arbitrary dimension, obtained by the procedure of strict hyperbolization. We also recognize these …
Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on hyperbolic 3-space.
Study on topological properties and boundaries of RCD(K,N) spaces.
problem Understanding the topology and boundaries of non-collapsed RCD(K,N) spaces.
method Established topological regularity and stability, introduced boundary concept.
result Properties of boundaries and behavior under convergence studied.
Recently, the old notion of causal boundary for a spacetime V has been redefined in a consistent way. The computation of this boundary ∂V for a standard conformally stationary spacetime V = R x M, suggests a natural compactification MB associated to any Riemannian metric on M or, more generally, to any Fin…
The paper shows how sublinearly Morse boundaries can be understood through combinatorial methods.
problem Understanding sublinearly Morse boundaries in cubulated groups and CAT(0) cube complexes.
method Combining geometric and combinatorial approaches to analyze sublinearly Morse boundaries.
result Sublinearly Morse boundaries can be described combinatorially and continuously related to Gromov and Roller boundaries.
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.
Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the co…
Random trees emerge from geodesics in hyperbolic groups.
problem Understanding geodesics in hyperbolic groups.
method Surveying known properties and constructing random trees.
result Rich random geometry of emerging trees.
Every Gromov hyperbolic group can be described by a finite subdivision rule.
problem Describing Gromov hyperbolic groups using subdivision rules.
method Finite subdivision rules acting on the 3-sphere to describe groups.
result Extends the representation of hyperbolic groups to non-cubulated groups.
We show that for a generic simple closed curve C in the asymptotic boundary of a Gromov hyperbolic 3-space with cocompact metric X, there exist a unique least area plane P in X with asymptotic boundary C. This result has interesting topological applications for constructions of canonical 2-dimensional objects in 3-mani…