Introduces hierarchical hyperbolic spaces for non-experts.
problem Understanding hierarchical hyperbolic spaces for non-experts.
method No specific method mentioned; aimed at non-experts.
result Introduces hierarchical hyperbolic spaces for non-experts.
Multicurve stabilizers' extensions are hierarchically hyperbolic.
problem Characterizing the structure of multicurve stabilizers' extensions.
method Proving the extensions of multicurve stabilizers are hierarchically hyperbolic groups.
result Extensions of multicurve stabilizers are hierarchically hyperbolic.
We give several sufficient conditions for uniform exponential growth in the setting of virtually torsion-free hierarchically hyperbolic groups. For example, any hierarchically hyperbolic group that is also acylindrically hyperbolic has uniform exponential growth. In addition, we provide a quasi-isometric characterizati…
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.
Boundary properties of hyperbolic groups are invariant under a maximization procedure.
problem Proving boundary properties of hierarchically hyperbolic groups are invariant.
method Proving boundary invariance under a maximization procedure.
result Boundary properties of hierarchically hyperbolic groups are invariant under maximization.
The study shows how quotients of mapping class groups are hierarchically hyperbolic.
problem Understanding the hierarchical hyperbolicity of mapping class groups and their quotients.
method A combinatorial criterion for hierarchical hyperbolicity applied to mapping class groups.
result Quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic.
Classifies 3-manifold groups with equivariant hierarchically hyperbolic structures.
problem Classifying 3-manifold groups with equivariant hierarchically hyperbolic structures.
method Construction of suitable quasimorphisms on Seifert pieces to construct actions on quasi-lines.
result 3-manifold groups admit equivariant hierarchically hyperbolic structures.
New groups are hyperbolic and rigid in mapping class groups.
problem Understanding the structure of Veech groups in mapping class groups.
method Showed that Veech groups are hierarchically hyperbolic and quasi-isometrically rigid.
result Veech groups are hierarchically hyperbolic and quasi-isometrically rigid.
Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichmüller space, right-angled Artin groups, and many other cubical groups. Given such a space X, we build a bordificationcompatible with the hierarchically hyperbolic structure. If $\mathc…
Outer automorphism group of hyperbolic groups is HHG under certain conditions.
problem Characterizing the outer automorphism group of hyperbolic groups.
method Proving finite-index subgroups are central extensions of orbifold mapping class groups with bounded Euler class.
result Outer automorphism group of a one-ended hyperbolic group is virtually a hierarchically hyperbolic group.
Uniform undistortion in cyclic subgroups of certain groups.
problem Understanding undistorted subgroups in group actions.
method Using quasimorphisms and hierarchically hyperbolic groups.
result Sharp examples of undistorted subgroups in hierarchically hyperbolic groups.
New group not biautomatic, geometrically constructed.
problem Constructing a non-biautomatic group.
method Using a CAT(0) hierarchically hyperbolic group and geodesic currents.
result First example of a non-biautomatic CAT(0) group of flat-rank 2.
New combinatorial structure for hierarchically hyperbolic spaces.
problem Constructing new hierarchically hyperbolic spaces.
method Combinatorial hierarchical hyperbolicity criterion to construct and clarify HHS structures.
result HHSs admit a combinatorial structure, clarifying the application of the combinatorial HHS criterion.
The study shows pseudo-Anosovs are common in mapping class groups.
problem Counting pseudo-Anosovs in mapping class groups.
method Using weakly contracting isometries and Morse elements.
result Pseudo-Anosovs are generic in mapping class groups.
We consider two manifestations of non-positive curvature: acylindrical actions on hyperbolic spaces and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for studying many important families of groups, including mapping class groups, right…
Study of Dehn filling quotients in hierarchically hyperbolic groups.
problem Understanding the structure of Dehn filling quotients in specific groups.
method Introduced a construction for cusped spaces of relatively hyperbolic groups and used it to study Dehn-filling-like quotients.
result Infinite hyperbolic quotients of mapping class groups of punctured spheres and braid groups are found.
We prove that all hierarchically hyperbolic spaces have finite asymptotic dimension and obtain strong bounds on these dimensions. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the mapping class group of a finite type surface: improving the bound from exponential to …
Logarithmic separation profile in hyperbolic groups shows hierarchical structure.
problem Understanding hierarchical structure in hyperbolic groups with logarithmic separation.
method Proving groups with logarithmic separation split over cyclic groups and providing counterexamples.
result Not all groups with hierarchical structure have logarithmic separation profile.
Hierarchically hyperbolic spaces (HHSs) are a large class of spaces that provide a unified framework for studying the mapping class group, right-angled Artin and Coxeter groups, and many 3--manifold groups. We investigate strongly quasiconvex subsets in this class and characterize them in terms of their contracting pro…
This paper generalizes property (QT) to a broader class of groups.
problem Proving property (QT) for a wider range of groups.
method Using projection complex machinery and hierarchical hyperbolic groups.
result Established sufficient conditions for groups to have property (QT).
In the context of CAT(0) cubical groups, we develop an analogue of the theory of curve complexes and subsurface projections. The role of the subsurfaces is played by a collection of convex subcomplexes called a \emph{factor system}, and the role of the curve graph is played by the \emph{contact graph}. There are a numb…
Extensions of Veech groups using hierarchical hyperbolic spaces.
problem Generalizing Veech groups to non-lattice cases.
method Using a hyperbolic space E^ and a nice action of Γ on it. result Finitely generated Veech groups are hierarchically hyperbolic.
We provide negative answers to questions posed by Durham, Hagen, and Sisto on the existence of boundary maps for some hierarchically hyperbolic spaces, namely maps from right-angled Artin groups to mapping class groups. We also prove results on existence of boundary maps for free subgroups of mapping class groups.
Stable subgroups identified in genus two handlebody group.
problem Characterizing stable subgroups in genus two handlebody group.
method Proving genus two handlebody group is hierarchically hyperbolic, using quasi-isometric embedding properties and Hamenstädt-Hensel construction.
result Stable subgroups identified and characterized.
We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our c…
Stable cubulations and bicombings in mapping class groups and Teichmüller spaces.
problem Understanding geometric structures in mapping class groups and Teichmüller spaces.
method Proving stably approximated by CAT(0) cube complexes, applying to broader colorable hierarchically hyperbolic spaces and groups.
result Stable cubulations and bicombings in mapping class groups and Teichmüller spaces, with stable coarse barycenters.
Characterizes boundaries of HHGs and their properties.
problem Understanding boundaries and properties of HHGs.
method Simplicial structure on hierarchically hyperbolic boundaries, modifications of HHG structures.
result Characterizes relative hyperbolicity and thickness of HHGs.
We generalize a result of Paulin on the Gromov boundary of hyperbolic groups to the Morse boundary of proper, maximal hierarchically hyperbolic spaces admitting cocompact group actions by isometries. Namely we show that if the Morse boundaries of two such spaces each contain at least three points, then the spaces are q…
Survey discusses new ideas in geometric group theory and their applications.
problem Understanding geodesic metric spaces and their equivariant wall structures.
method Introduces and highlights the impact of injective metric spaces and cubical approximation theorem.
result Rich equivariant wall structures in various geodesic metric spaces.
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…
The paper provides conditions for amalgamation of certain subgroups and preserves convexity properties.
problem Conditions for amalgamation of subgroups in hierarchically hyperbolic groups.
method Study of amalgamation conditions and preservation of convexity properties.
result Conditions under which amalgamation preserves hierarchical quasiconvexity and strong quasiconvexity.
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.
We prove a geometric model for HHS hierarchies as CAT(0) cube complexes.
problem Modeling hierarchically hyperbolic spaces as cube complexes.
method Prove quasi-median quasi-isometry between hulls and cubical models.
result Hierarchical boundaries are locally modeled by CAT(0) cube complexes.
Study on embedding tree products into groups, distinguishing them.
problem Quasi-isometric embedding of tree products into various groups.
method Using coarse embeddings of products of bushy trees into hierarchically hyperbolic spaces.
result Quasi-isometrically distinguish and rule out embeddings between groups.
Random quotients of mapping class groups have rigid properties.
problem Rigidity of random quotients of mapping class groups.
method Generalization of Ivanov's theorem and use of hierarchically hyperbolic groups.
result Automorphisms and commensurators of random quotients coincide with the groups themselves.
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.
In this paper, we establish upper bounds on the length of the shortest conjugator between pairs of infinite order elements in a wide class of groups. We obtain a general result which applies to all hierarchically hyperbolic groups, a class which includes mapping class groups, right-angled Artin groups, Burger--Mozes-ty…
The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Examples for which the rank coincides with familiar quantities include: the dimension of maximal Dehn twi…
The paper studies a group action on a hyperbolic space derived from a lattice Veech group.
problem Investigating the geometry of a Veech group and its extensions.
method Analyzing the fundamental group of a bundle with singular Euclidean-by-hyperbolic geometry, collapsing regions to produce a hyperbolic action.
result The Veech group's fundamental group acts on a hyperbolic space, retaining most of its geometry.
Develops geometric foundations for sublinear Morse boundaries in mapping class groups and Teichmüller spaces.
problem Capturing generic directions in mapping class groups and Teichmüller spaces.
method Develops tools for modeling hulls of median rays in hierarchically hyperbolic spaces via CAT(0) cube complexes.
result Sublinear Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the curve graph.
Survey of tools for studying hierarchical hyperbolic spaces.
problem Understanding hierarchical hyperbolic spaces.
method Various tools developed for studying HHSs.
result Ease of use for non-experts in HHS machinery.
New bicombings found for mapping class groups and Teichmüller spaces.
problem Finding efficient ways to navigate mapping class groups and Teichmüller spaces.
method Explained bicombings via stable cubical intervals in hierarchically hyperbolic spaces.
result Hierarchical hulls are quasi-isometric to finite CAT(0) cube complexes.
Let X be a proper CAT(0) cube complex admitting a proper cocompact action by a group G. We give three conditions on the action, any one of which ensures that X has a factor system in the sense of [BHS14]. We also prove that one of these conditions is necessary. This combines with results of Behrstock--Hagen--Sisto to s…
Hyperbolic space outperforms Euclidean in learning hierarchical data.
problem Learning hierarchical data in Euclidean space requires exponentially many samples.
method Established geometric obstruction in Euclidean space and showed hyperbolic space's advantage.
result Hyperbolic space enables learning with O(mRlogm) samples, matching information-theoretic optimum. New topology shows Morse boundaries are topologically invariant.
problem Topological invariance of Morse boundaries in CAT(0) cubical groups.
method New hyperbolic topology induced by group actions on hyperbolic spaces.
result Sublinearly Morse boundaries are homeomorphic up to visual topology.
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.
A new drug embedding method using hierarchical drug relations and chemical structures.
problem Learning accurate drug representations from chemical structures and hierarchies.
method Semi-supervised drug embedding using VAE in hyperbolic space.
result The method accurately places drugs in a hierarchy and predicts side-effects.
Paper provides first theoretical guarantees for hyperbolic space learning.
problem Learning a classifier in hyperbolic space for hierarchical data.
method Efficient algorithm for large-margin hyperplane learning in hyperbolic space.
result The low embedding dimension in hyperbolic space leads to superior classifier learning guarantees.