Survey on quasi-isometries of group pairs and their invariants.
problem Understanding quasi-isometries of group pairs and their invariants.
method Exploration of quasi-isometry and qi-characteristic collections of subgroups.
result New insights into phenomena observed in quasi-isometric rigidity.
Study quasi-isometry invariants of square complexes and their applications.
problem Classifying quasi-isometry types of 2D right-angled Artin groups and graph 2-braid groups.
method Define and analyze intersection complexes for universal covers of weakly special square complexes.
result Discover new quasi-isometric relationships between graph 2-braid groups and right-angled Artin groups.
Bowditch's JSJ tree for splittings over 2-ended subgroups is a quasi-isometry invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian. Our main result gives an explicit, computable "visual" construction of this tree for certain hyperbolic right-angled Coxeter groups. As an application of our constructi…
New concept SB-generation helps classify transformation groups.
problem Classifying transformation groups through quasi-isometry invariants.
method Identifying SB-generated groups in specific transformation groups.
result SB-generation provides robust extension of finite generation.
New metric space cohomology relates to Riemannian manifold cohomology.
problem Understanding cohomology structures on Riemannian and contact manifolds.
method Relating Lq,p-cohomology to ℓq,p-cohomology and proving quasi-isometry invariance. result Quasi-isometry invariance and multiplicative structure of Lq,p-cohomology. Complex captures group properties, invariant under quasi-isometry.
problem Classical properties of subgroups in a group pair.
method Introduces coset intersection complex to study group properties.
result Quasi-isometry invariance of coset intersection complex.
The paper proves that relative Dehn functions are invariant under quasi-isometry.
problem Invariance of relative Dehn functions under quasi-isometry.
method Proof of quasi-isometry invariance of relative Dehn functions.
result Relative Dehn functions are invariant under quasi-isometry.
New invariant classifies right-angled Coxeter groups, bounds thickness.
problem Classifying and bounding right-angled Coxeter groups.
method Introducing hypergraph index from defining graph, computing upper bounds.
result Hypergraph index partitions groups into quasi-isometry classes, bounds thickness.
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. The paper introduces new invariants to detect hyperbolic parts in relatively hyperbolic groups.
problem Detecting hyperbolic parts in relatively hyperbolic groups.
method Quasi-isometry invariants based on small cancellation theory over free products.
result Infinitely many quasi-isometry types of one-ended hyperbolic relative groups can be constructed.
The paper explores additional structures on Morse boundaries to distinguish hyperbolic spaces up to quasi-isometry.
problem Distinguishing hyperbolic spaces up to quasi-isometry using additional structures on Morse boundaries.
method Investigates additional structures on Morse boundaries and proves conditions for a homeomorphism to be induced by a quasi-isometry.
result A homeomorphism between Morse boundaries of hyperbolic spaces is induced by a quasi-isometry if and only if it is bihölder, quasi-symmetric, or strongly quasi-conformal.
Introduces Morse quasiflats and proves their equivalence and quasi-isometry invariance.
problem Generalizing Morse quasigeodesics to arbitrary dimensions.
method Introduces alternative definitions and proves their equivalence under appropriate assumptions.
result Morse quasiflats are asymptotically conical and have canonically defined Tits boundaries.
Measure-scaling quasi-isometries on graphs have specific scaling groups.
problem Understanding the scaling groups of graphs under quasi-isometries.
method Analyzing measure-scaling quasi-isometries on graphs and their properties.
result The scaling group of a graph is invariant under measure-scaling quasi-isometries.
Classifies certain graph 2-braid groups up to quasi-isometry.
problem Classifying 2-braid groups over graphs up to quasi-isometry.
method Using intersection complexes and right-angled Artin groups.
result Classifies 2-braid groups over graphs with circumference ≤ 1 up to quasi-isometry.
Quasi-isometries in horospherical products are close to product maps.
problem Understanding the rigidity of quasi-isometries in specific geometric spaces.
method Proving quasi-isometries are uniformly close to product maps in horospherical products of hyperbolic spaces.
result Quasi-isometries of horospherical products of hyperbolic spaces are geometrically rigid.
The study characterizes elements of a group quotient and finds a non-locally indicable left-orderable group.
problem Understanding the structure and properties of a specific group quotient.
method Introduced an invariant for quasi-isometries of the positive real line and split it into units.
result Found a quotient of the quasi-isometry group of the positive real line that is left-orderable but not locally indicable.
As demonstrated by Croke and Kleiner, the visual boundary of a CAT(0) group is not well-defined since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. We introduce a new type of boundary for a CAT(0) space, called the contracting boundary, made up rays satisfying one of five hyperbolic-like propertie…
In this paper we construct and study isoperimetric functions at infinity for Hadamard manifolds. These quasi-isometry invariants give a measure of the spread of geodesics in such a manifold.
New approach uses distributed persistence for stable, parallelizable topological analysis of large point clouds.
problem Estimating the full persistence diagram of large point clouds is expensive, unstable, and not a sufficient statistic.
method Proposes distributed persistence as a new invariant, which is perfectly parallelizable, more stable, and has a rich inverse theory.
result The map from point clouds to distributed persistence invariants is a global quasi-isometry, interpolating between purely geometric and topological invariants.
Introduces halo products and studies their geometric properties.
problem Understanding the large-scale geometry of halo groups.
method Introduces halo products and builds a geometric framework.
result Provides refined invariants distinguishing halo groups up to quasi-isometry.
This study extends a result on quasi-isometry of hyperbolic groups to relatively hyperbolic groups.
problem Classifying groups up to quasi-isometry, focusing on relatively hyperbolic groups.
method Defining quasiconformal maps and showing their equivalence to coarsely cusp-preserving quasi-isometries between Bowditch boundaries.
result Quasiconformal maps between Bowditch boundaries of relatively hyperbolic groups are equivalent to coarsely cusp-preserving quasi-isometries.
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.
A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric nilpotent groups. The nonsurjective injection defining such an extension induces…
Study L2-cohomology in unbounded geometry manifolds.
problem Invariance of L2-cohomology under quasi-isometries on unbounded ends. method Uniform homotopy equivalence, quasi-isometry on unbounded ends, mapping cone for L2-cohomology. result Invariance of L2-cohomology groups under quasi-isometry on unbounded ends. The study describes quasiflats in 2D Artin groups and their properties.
problem Understanding the structure and properties of quasiflats in 2D Artin groups.
method Metric systolicity and combinatorial analysis of tilings.
result Precise description of building blocks (atomic sectors) for quasiflats in 2D Artin groups.
In this paper it is proved that relative hyperbolicity is an invariant of quasi-isometry. As a byproduct of the arguments, simplified definitions of relative hyperbolicity are obtained. In particular we obtain a new definition very similar to the one of hyperbolicity, relying on the existence for every quasi-geodesic t…
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.
The paper defines quasi-isometry for almost contact metric manifolds and explores its properties.
problem Understanding quasi-isometry in almost contact metric manifolds.
method Definition and study of quasi-isometry for almost contact metric manifolds.
result Established a relation between scalar curvature and quasi-isometric constants.
New research shows uncountably many quasi-isometry classes of groups of type FP.
problem Identifying distinct quasi-isometry classes of groups of type FP. method Constructing uncountable families of groups and proving quasi-isometry classes.
result Uncountably many quasi-isometry classes of groups of type FP. The abstract discusses the classification of 3D Lie groups with Riemannian metrics.
problem Classifying 3D Lie groups up to quasi-isometries and bi-Lipschitz equivalence.
method Review of existing literature and study of quasi-isometry and bi-Lipschitz equivalence.
result For three-dimensional simply connected groups, quasi-isometry implies isometry with suitable metrics.
New metric space concept and quasi-isometry properties explored.
problem Exploring new metric spaces and quasi-isometry properties.
method Introducing (b,c)-metric and defining collapsing maps.
result Collapsing maps preserve quasi-isometry properties.
Study introduces combinatorial criterion for quasi-isometry groups of Euclidean spaces.
problem Determining quasi-isometries of Euclidean spaces.
method Introduces PLδ-homeomorphisms and combinatorial criterion using vertices and edges of simplicial structures. result The center of the quasi-isometry group QI(Rn) is trivial. Study confirms optimal bounds for group cohomology of Lie groups.
problem Optimal bounds for group cohomology of Lie groups.
method Combining complementary vanishings with spectral sequences and quasi-isometry invariance.
result Non-vanishing of group Lp-cohomology for large p and equal degree to rank. The study classifies Heintze groups up to isometry and quasi-isometry in low dimensions.
problem Classifying Heintze groups up to isometry and quasi-isometry in low dimensions.
method Analyzing quasi-isometries and isometries of Heintze groups, applying existing tools to groups of dimension 4 and 5.
result Complete classification of simply connected solvable groups in dimension 4 and groups of polynomial growth in dimension 5 up to isometry.
This paper overviews recent developments in the classification up to quasi-isometry of finitely generated groups, and more specifically of relatively hyperbolic groups.
This paper explores when homeomorphisms between Morse boundaries of spaces induce quasi-isometries.
problem When does a homeomorphism between Morse boundaries of spaces imply a quasi-isometry?
method Investigates quasi-mobius homeomorphisms and 2-stability conditions.
result A homeomorphism between Morse boundaries of proper, cocompact spaces is induced by a quasi-isometry if and only if it is quasi-mobius and 2-stable.
A special group of transformations of the real line cannot act effectively on it.
problem Understanding the limitations of transformations on the real line.
method Analyzing the group of orientation-preserving quasi-isometries of the real line.
result The group of quasi-isometries of the real line cannot act effectively on the line.
Generalizes Bestvina's Z-boundaries to coarse Z-boundaries.
problem Establishing properties of Z-boundaries for groups. method Introducing a new concept of a 'coarse Z-boundary' and proving theorems about it. result Admitting a coarse Z-boundary is a pure quasi-isometry invariant. The center of the group of quasi-isometries of the real line is trivial.
problem Identifying the center of the group of quasi-isometries of the real line.
method Analyzing the group structure and using the quasi-isometry properties to show the triviality of the center.
result The center of the group of quasi-isometries of the real line is trivial.
Any quasi-isometry of the complex of curves is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.
De Rham theorem extended to Orlicz cohomology.
problem Extending de Rham's theorem to a broader class of cohomology.
method Proving isomorphism between de Rham Lφ-cohomology and simplicial ℓφ-cohomology. result Isomorphism between de Rham Lφ-cohomology and simplicial ℓφ-cohomology. New surface without quasi-isometric triangulations found.
problem Existence of quasi-isometric triangulations on surfaces.
method Constructing a complete Riemannian surface with specific properties.
result Found a surface without any triangulation quasi-isometric to the surface.
We give a method of constructing maps between tubular groups inductively according to a set of strategies. This map will be a quasi-isometry exactly when the set of strategies is consistent. Conversely, if there exists a quasi-isometry between tubular groups, then there is a consistent set of strategies for them. There…
We prove that PSL(2,Z[1/p]) gives the first example of groups which are not quasi-isometric to each other but have the same quasi-isometry group. Namely, PSL(2,Z[1/p]) and PSL(2,Z[1/q]) are not quasi-isometric unless p=q, and, independent of p, the quasi-isometry group of PSL(2,Z[1/p]) is PSL(2,Q). In addition, we char…
In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to R⋉Rn wh…
Similarity found in metrics on special Lie groups.
problem Comparing Riemannian metrics on specific Lie groups.
method Proved all metrics are roughly similar via identity.
result All left-invariant Riemannian metrics are roughly similar.
New classes of RAAGs have quasi-isometry coinciding with commensurability.
problem Quasi-isometry classification of RAAGs with infinite outer automorphism groups.
method Deformation argument and cubulation techniques.
result For certain RAAGs, quasi-isometry implies commensurability.
In this paper we study a homological version of the higher-dimensional divergence invariants defined by Brady and Farb. We show that they are quasi-isometry invariants in the class of proper cocompact Hadamard spaces in the sense of Alexandrov and that they can moreover be used to detect the Euclidean rank of such spac…