Study abelian subgroups in 3-manifolds, linking to splitting properties.
problem Separation of abelian peripheral subgroups under conjugacy and malnormality.
method Examining acylindrical splittings of 3-manifold groups.
result Existence of acylindrical splittings related to abelian subgroups.
Explicit bounds on group size for certain geometric actions.
problem Bounding group size in geometric actions with bounded entropy.
method Proving an explicit function F(k, E) for groups with k-acylindrical splittings.
result Groups with bounded entropy have a finite size bound.
Study shows L2-Betti numbers vanish for certain matrix groups over rings, proving non-acylindrical hyperbolicity.
problem Investigating L2-Betti numbers and acylindrical hyperbolicity for matrix groups over various rings. method Utilizing n-rigid rings, the study proves vanishing L2-Betti numbers and non-acylindrical hyperbolicity for specified groups. result Matrix groups over certain rings have vanishing L2-Betti numbers and are not acylindrically hyperbolic. We present an algorithm which given a presentation of a group G without 2-torsion, a solution to the word problem with respect to this presentation, and an acylindricity constant κ, outputs a collection of tracks in an appropriate presentation complex. We give two applications: the first is an algorithm which decid…
The study examines Heegaard splittings defined by Dehn twists and finds hyperbolic metrics with specific geodesic lengths.
problem Characterizing Heegaard splittings defined by Dehn twists and their geometric properties.
method Examining Heegaard splittings of genus g≥3 defined by the n-th power of a Dehn twist along a pared acylindrical curve. result For n≥14, the Heegaard splitting has a hyperbolic metric with a closed geodesic of length between 0.7/(n2g2) and 34.3/n2. Three versions of the Freiheitssatz are proved in the context of one-relator quotients of limit groups, where the latter are equipped with 1-acylindrical splittings over cyclic subgroups. These are natural extensions of previously published corresponding statements for one-relator quotients of orientable surface groups…
The paper studies acylindrical actions on trees and proves acylindrical hyperbolicity of Baumslag-Solitar groups.
problem Exploring acylindrical actions on trees and their properties.
method Demonstrates criteria for preserving acylindrical hyperbolicity and analyzes the outer automorphism group of Baumsligar-Solitar groups.
result Proves acylindrical hyperbolicity of non-solvable Baumsligar-Solitar groups.
New groups defined that act on trees without repeating.
problem Understanding groups acting on trees without repeating.
method Developed a new concept of acylindrically arboreal groups and classified specific groups.
result Provided examples of groups with tree actions but no non-elementary acylindrical actions.
Proves WPD elements are exponentially generic in acylindrically hyperbolic groups.
problem Growth rates of acylindrically hyperbolic groups.
method Analyzes exponential growth rates of non-WPD and WPD elements.
result WPD elements are exponentially generic in acylindrically hyperbolic groups.
Automorphism groups of infinitely-ended groups are acylindrically hyperbolic.
problem Characterizing the hyperbolicity of automorphism groups of infinitely-ended groups.
method Proving acylindrical hyperbolicity through group properties and automorphism analysis.
result Automorphism groups of infinitely-ended groups are acylindrically hyperbolic.
Study shows Morse elements are common in acylindrically hyperbolic groups.
problem Understanding generic elements in acylindrically hyperbolic groups.
method Analyzing Morse elements and outer automorphisms.
result Morse elements are common in acylindrically hyperbolic groups.
Simplified projection complexes for acylindrical actions proved.
problem Proving acylindrical actions on projection complexes.
method Introduced a sharper Behrstock inequality and used it to simplify projection complexes.
result Proved acylindrical actions on simplified projection complexes.
We give a criterion to prove that some groups are not acylindrically hyperbolic. As an application, we prove that the mapping class group of an infinite type surface is not acylindrically hyperbolic.
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.
The Farrell-Jones Conjecture holds for groups acting acylindrically on trees.
problem Verifying the Farrell-Jones Conjecture for groups acting on trees.
method Analyzing acylindrical actions on simplicial trees and using the Farrell-Jones Conjecture.
result The Farrell-Jones Conjecture holds for groups acting acylindrically on trees.
New group Q has unusual properties like no uniform non-amenability.
problem Properties of acylindrically hyperbolic groups.
method Proving a common quotient and applying to a specific group.
result Found a group Q with strong fixed point properties and unusual characteristics. Study projection in acylindrically hyperbolic groups, proving sublinear tracking and growth bounds.
problem Projection phenomena in acylindrically hyperbolic groups.
method Analyzing shortest projections in word metrics and hyperbolic spaces.
result Sublinear tracking of shortest projections and effective growth bounds.
Random quotients preserve hyperbolic properties in groups.
problem Preserving hyperbolic properties in random group quotients.
method Independent random walks, spinning families, projection complexes.
result Random quotients of acylindrical and hierarchical hyperbolic groups remain so.
We prove that every finitely presented group with positive first ℓ2-Betti number that virtually surjects onto Z is acylindrically hyperbolic. In particular, this implies acylindrical hyperbolicity of finitely presented residually finite groups with positive first ℓ2-Betti number as well as groups …
We provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees. In particular, we consider amalgamated products and HNN-extensions, 1-relator groups, automorphism groups of polynomial algebras, 3-manifold groups and graph products. Acylindrical hyperbolicity is then used to obtain …
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.
We show that the verbal width is infinite for acylindrically hyperbolic groups, which include hyperbolic groups, mapping class groups and Out(Fn).
The class of acylindrically hyperbolic groups, which are groups that admit a certain type of non-elementary action on a hyperbolic space, contains many interesting groups such as non-exceptional mapping class groups and Out(Fn) for n≥2. In such a group, a generalized loxodromic element i…
Artin groups not free of infinity are shown to have finite centers.
problem Characterizing Artin groups with finite centers.
method Reduced clique-cube complexes and actions on them.
result Artin groups not free of infinity have finite centers, and are trivial in many cases.
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.
New largest acylindrical actions found for hierarchically hyperbolic groups.
problem Understanding non-positive curvature in hierarchically hyperbolic groups.
method Study of acylindrical actions and quasigeodesic stability in hierarchically hyperbolic groups.
result Hierarchically hyperbolic groups have a unique largest acylindrical action.
We show that the aspherical manifolds produced via the relative strict hyperbolization of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automor…
We prove an acylindrical accessibility theorem for finitely generated groups acting on R-trees. Namely, we show that if G is a freely indecomposable non-cyclic k-generated group acting minimally and M-acylindrically on an R-tree X then for any ε>0 there is a finite subtree Yε⊆X…
Study shows infinite dimensional zero norm subspace in bounded cohomology of acylindrically hyperbolic groups.
problem Understanding the zero norm subspace in bounded cohomology of acylindrically hyperbolic groups.
method Introduced combinatorial volume forms and a new seminorm on exact bounded cohomology to construct non-trivial classes.
result Shows infinite dimensional zero norm subspace in degree 3 bounded cohomology of acylindrically hyperbolic groups.
An exotic plane exists in an acylindrical 3-manifold without being closed.
problem Does a geodesic plane in an acylindrical 3-manifold remain closed or dense?
method Explicit construction and analysis of a specific example.
result An example of a geodesic plane that is closed in the interior but not in the manifold itself.
Formula found for skinning map contraction in hyperbolic geometry.
problem Finding the contraction constant of skinning maps.
method Elementary hyperbolic geometry and moduli spaces of hyperbolic manifolds.
result Explicit formula for contraction constant.
Survey of recent Kleinian representation convergence results.
problem Kleinian representation convergence
method Survey and analysis of recent results following Thurston's theorems
result Survey of recent and less recent results on convergence of Kleinian representations
Random subgroups of hyperbolic groups often form free groups and are hyperbolically embedded.
problem Understanding the structure of random subgroups in acylindrically hyperbolic groups.
method Examining a random subgroup generated by independent random walks.
result Random subgroups are often free groups and hyperbolically embedded in the group.
We prove that the outer automorphism group Out(G) is residually finite when the group G is virtually compact special (in the sense of Haglund and Wise) or when G is isomorphic to the fundamental group of some compact 3-manifold. To prove these results we characterize commensurating endomorphisms of acylindrical…
Random walks on hyperbolic spaces show linear progress with exponential decay.
problem Understanding progress and decay in random walks on hyperbolic spaces.
method Analyzing random walks on separable, geodesic hyperbolic metric spaces with specific step distributions.
result Exponential decay in progress is extended to non-acylindrical actions.
Extra large Artin groups are simple and have unique geodesics.
problem Characterizing and understanding XXL type Artin groups.
method Simple locally CAT(0) classifying space and rank 1 periodic geodesic.
result Extra large type Artin groups are acylindrically hyperbolic.
The aim of this note is to give the simplest possible proof that Mapping Class Groups of closed hyperbolic surfaces are acylindrically hyperbolic, and more specifically that their curve graphs are hyperbolic and that pseudo-Anosovs act on them as loxodromic WPDs.
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.
The paper proves properties of geodesic planes in a specific type of 3-manifold.
problem Characterizing geodesic planes in a specific class of 3-manifolds.
method Analyzing geodesic planes within the convex core of a hyperbolic 3-manifold.
result Geodesic planes are either closed or dense, with only countably many being closed.
Characterizes contracting isometries in CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups.
problem Characterizing contracting isometries in CAT(0) cube complexes and their relation to acylindrical hyperbolicity.
method Characterization of contracting isometries without local finiteness assumption, combinatorial boundary introduction, and application to diagram groups.
result Determine precise conditions for acylindrical hyperbolicity of diagram groups.
Improved bounds on acylindricity for right-angled Artin groups.
problem Bounding the acylindrical action of right-angled Artin groups on their extension graphs.
method Exploring lattice properties, studying prefixes of powers, and extending quasi-root uniqueness.
result Cardinality of r-quasi-stabilizer is bounded by a linear function of r. The study bounds curve sizes on surfaces using group cubulations.
problem Bounding curves on surfaces.
method Group cubulations and acylindrical accessibility.
result Bounds on curves on surfaces.
We consider closed acylindrical surfaces in 3-manifolds and in knot and link complements, and show that the genus of these surfaces is bounded linearly by the number of tetrahedra in the triangulation of the manifold and by the number of rational (or alternating) tangles in a projection of a link (or knot). For each g …
The study shows rigidity properties are lost in hyperbolic generalizations.
problem Lack of geometric and topological rigidity in hyperbolic groups.
method Observations and proofs on acylindrically hyperbolic and relatively hyperbolic groups.
result No well-defined limit set for acylindrical actions on hyperbolic spaces.
Study on Cheeger constant in specific hyperbolic 3-manifolds.
problem Analyzing Cheeger constant in convex co-compact hyperbolic 3-manifolds.
method Examining Cheeger constant as a functional in the space of specific hyperbolic 3-manifolds.
result Global maximum of Cheeger constant is uniquely attained at the Fuchsian locus.
Bound skinning map diameter on hyperbolic 3-manifold rays.
problem Bounding skinning map diameter on hyperbolic 3-manifold rays.
method Using acylindrical hyperbolic 3-manifolds and thick Teichmüller geodesic rays.
result Diameter of skinning map bounded by a constant.
Groups without non-trivial finite normal subgroups have a strong ping pong property.
problem Proving a strong ping pong property for acylindrically hyperbolic groups.
method Proving the Pnaive property for groups with no non-trivial finite normal subgroups. result Groups satisfying the condition have a strong ping pong property.
Cohomology defines hyperbolic spaces and their subgraphs.
problem Characterizing hyperbolic spaces and their subgraphs.
method Complete cohomological characterization using ℓ∞-cohomology. result Cohomology vanishing characterizes hyperbolicity and acylindrical hyperbolicity.