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.
Extended Otal-Peigné's Theorem to Gromov-hyperbolic spaces.
problem Prove critical exponent equals topological entropy for group actions.
method Extended Otal-Peigné's Theorem to proper, Gromov-hyperbolic spaces.
result Critical exponent equals topological entropy for line-convex spaces.
We show that the number of twisted conjugacy classes is infinite for any automorphism of non-elementary, Gromov hyperbolic group . An analog of Selberg theory for twisted conjugacy classes is proposed.
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.
We study the variety of actions of a fixed (Chevalley) group on arbitrary geodesic, Gromov hyperbolic spaces. In high rank we obtain a complete classification. In rank one, we obtain some partial results and give a conjectural picture.
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.
Proves equivalence of two types of representations of free groups in hyperbolic spaces.
problem Equivalence of two types of representations of free groups in hyperbolic spaces.
method Independent proof of equivalence for free groups of rank two in Gromov-hyperbolic spaces.
result Set of Bowditch representations equals set of primitive-stable representations.
Paper proves vanishing homology groups for certain hyperbolic groups.
problem Understanding homology groups of specific hyperbolic groups.
method Using twisted Wirtinger presentations to prove homology group vanishing.
result Second homology groups vanish for certain Gromov hyperbolic groups.
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.
Compact manifolds with specific cover properties are hyperbolic.
problem Understanding Gromov hyperbolicity in compact manifolds.
method Proving Gromov hyperbolicity through coboundary expansion in residual covers.
result Compact manifolds with certain cover properties have hyperbolic fundamental groups.
Study on representations of four-punctured sphere group in hyperbolic spaces.
problem Understanding representations of the four-punctured sphere group.
method Investigation into simple-stable and Bowditch representations in Gromov-hyperbolic spaces.
result Simple-stable representations and Bowditch representations are equivalent.
New examples of 5D manifolds without certain structures.
problem Finding manifolds without specific geometric structures.
method Relative strict hyperbolization.
result Construction of 5D manifolds without real projective or flat conformal structures.
Minimal action on universal circle for foliations on 3-manifolds.
problem Action of fundamental group on universal circle of foliations.
method Analyzes uniform foliations on 3-manifolds, proving minimality and transitivity.
result Action is minimal and transitive on pairs of different points.
Paper proves subelliptic estimates for Kähler-Einstein metrics on convex domains.
problem Proving subelliptic estimates for Kähler-Einstein metrics on convex domains.
method Proved subelliptic estimates by constructing bounded plurisubharmonic functions.
result Subelliptic estimates for Kähler-Einstein metrics on convex domains.
New theorem on critical exponents for group actions.
problem Critical exponents of group actions.
method Proving critical exponents coincide under co-amenability condition.
result Generalizes previous results on critical exponents.
Classifies mapping tori of specific groups, generalizing known results.
problem Classifying mapping tori of specific groups.
method Using Hopf-type properties and Poincaré Duality groups.
result Generalizes and provides new proofs for fibered 3-manifolds.
The paper characterizes geometric infiniteness for convergence group actions using orbit uniform metrics.
problem Characterizing geometric infiniteness for convergence group actions.
method Introducing orbit uniform metrics and proving properties of discrete orbits.
result Characterization of geometric infiniteness in terms of uncountability of non-conical limit points and existence of escaping sequences.
Study of groups and their quasi-isometrically embedded subgroups.
problem Understanding the structure and properties of groups and their subgroups.
method Abstracting the notion of A/QI triples and using methods from geometric group theory.
result Stability of quasi-isometrically embedded subgroups in finitely generated groups.
Acylindrical hyperbolicity proven for a specific group of automorphisms.
problem Proving the acylindrical hyperbolicity of a group of automorphisms.
method Study of a 2D simplicial complex and proving its contractibility and Gromov-hyperbolicity; finding loxodromic elements.
result Tame automorphism group is acylindrically hyperbolic.
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.
Study examines large deviations in random walks on hyperbolic spaces.
problem Large deviations in random walks on Gromov-hyperbolic spaces.
method Established large deviations results for distance and translation length of random walks.
result Deduced a special case of a conjecture regarding spectral radii of random matrix products.
Random walks on hyperbolic spaces show linear growth in translation lengths.
problem Investigate the growth of translation lengths in random walks on hyperbolic spaces.
method Prove linear growth without moment conditions and apply to Teichmüller spaces.
result Linear growth of translation lengths in random walks on hyperbolic spaces.
Geodesic currents on hyperbolic surfaces have dual spaces that are metric trees.
problem Understanding the dual spaces of geodesic currents on hyperbolic surfaces.
method Analyzing the geometric properties of dual spaces, including their hyperbolicity and completeness.
result The dual spaces of geodesic currents are Gromov hyperbolic metric tree-graded spaces.
Random branched covers of groups are homotopy equivalent to geometrically small cancellation complexes.
problem Understanding the topological properties of random branched covers of groups.
method Constructing a random model for branched covers and showing asymptotic homotopy equivalence to geometrically small cancellation complexes.
result The fundamental group of a random branched cover is Gromov hyperbolic and has small cohomological dimension.
New method classifies Heintze groups using Lp-cohomology.
problem Quasi-isometry classification of Heintze groups.
method Introducing relative Lp-cohomology and applying it to Heintze groups. result Explicit construction of non-zero relative Lp-cohomology classes. Example groups show outer automorphism groups can have lower cohomological dimension than expected.
problem Outer automorphism groups of certain groups have lower cohomological dimension than expected.
method Constructing specific groups to demonstrate the phenomenon.
result Outer automorphism groups can have cohomological dimension lower than their geometric dimension.
The paper examines random walks on metric spaces and finds commensurable subgroups.
problem Determining commensurable subgroups via stationary measures in metric spaces.
method Analyzing random walks on isometry groups of metric spaces with non-singular stationary measures.
result Subgroups generated by random walks are commensurable under mild conditions.
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.
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…
Study of Bowditch representations in hyperbolic spaces with implications for dynamics and recognition.
problem Characterizing and understanding representations of free groups into hyperbolic spaces.
method Generalization of Bowditch conditions, explicit constant Kδ for hyperbolicity, characterizations of representations. result Linear growth of lengths for primitive elements in Bowditch representations, new characterization of primitive-stable representations.
Simplified potential theory for complex geometric spaces.
problem Developing potential theory for Gromov hyperbolic manifolds.
method Streamlining arguments to use coarse geometro-analytic invariants.
result Derived useful details not found in original sources.
The study shows a finite number of groups acting on hyperbolic spaces with bounded entropy and compact quotient.
problem Finite number of groups acting on hyperbolic spaces with bounded entropy and compact quotient.
method Analyzing torsion-free groups acting by isometries on hyperbolic metric spaces with bounded entropy and compact quotient.
result The set of such groups is finite and can be estimated based on hyperbolicity constant, entropy, and quotient diameter.
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…
New insights into the geometry of flows on 3-manifolds.
problem Understanding the geometry of flows on 3-manifolds.
method Analyzing the action of pseudo-Anosov flows on Gromov-hyperbolic spaces.
result Genericity of non-periodic elements in the fundamental group.
The paper examines cone-offs of CAT(0) cube complexes and their geometric properties.
problem Characterizing the hyperbolicity of certain geometric structures.
method Study of cone-offs over combinatorially convex subcomplexes, focusing on Gromov-hyperbolicity.
result Direct proof of relative hyperbolicity for right-angled Coxeter groups and acylindrical hyperbolicity for specific quotients.
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.
We study quasi-isometry invariants of Gromov hyperbolic spaces, focussing on the l_p-cohomology and closely related invariants such as the conformal dimension, combinatorial modulus, and the Combinatorial Loewner Property. We give new constructions of continuous l_p-cohomology, thereby obtaining information about the l…
We prove that the first reduced cohomology with values in a mixing Lp-representation, p larger than 1, vanishes for a class of amenable groups including connected amenable Lie groups. In particular this solves for this class of amenable groups a conjecture of Gromov saying that every finitely generated amenable group h…
We prove that the boundary of a right-angled hyperbolic building is a universal Menger space. Corollary: the 3-dimensional universal Menger space is the boundary of some Gromov-hyperbolic group.
New examples of hyperbolic manifolds with convex projective structures.
problem Finding new examples of hyperbolic manifolds with convex projective structures.
method Generalized Dehn filling and real projective structures.
result Infinitely many hyperbolic manifolds with cusps admit properly convex real projective structures.
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.
Study critical exponents of invariant subgroups in hyperbolic spaces.
problem Understanding critical exponents of invariant subgroups in hyperbolic spaces.
method Defined critical exponent δ(μ) and used a maximal ergodic theorem for hyperbolic groups.
result Critical exponent δ(μ) > d/2 in general and δ(μ) = d for divergence type subgroups.
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. We use an accessibility result of Delzant and Potyagailo to prove Swarup's Strong Accessibility Conjecture for Gromov hyperbolic groups with no 2-torsion. It follows that, if M is an irreducible, orientable, compact 3-manifold with hyperbolic fundamental group, then any hierarchy in which M is decomposed alternately al…
If M is a compact oriented manifold-with-boundary whose fundamental group is virtually nilpotent or Gromov-hyperbolic, we show that the higher signatures of M are oriented-homotopy invariants.
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.
We give a necessary complex geometric condition for a bounded smooth convex domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic. More precisely, we prove that if a smooth bounded convex domain contains an analytic disk in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi di…
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.