A coarse invariantmath.MG
This note extends the invariant defined in "An invariant of metric spaces under bornologous equivalences" to the coarse category.
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48 results for “coarse invariant”
Study semi-coarse spaces' homotopy and homology, extending coarse geometry.
problem Extend homotopy and homology concepts to semi-coarse spaces.
method Analyze homotopy and construct homology groups invariant under semi-coarse homotopy equivalence.
result Show semi-coarse homology is isomorphic to Vietoris-Rips homology for graphs.
Bornological metrics on groups are studied, showing equivalence classes and constructing non-equivalent improper metrics.
problem Characterizing and constructing left-invariant metrics on groups that are not proper.
method Introducing bornological metrics and studying their equivalence classes, constructing non-equivalent improper metrics.
result Each coarse equivalence class of bornological metrics is determined by a bornology, and every class contains a canonical left-invariant representative.
We study the concept of coarse disjointness and large scale -to- functions. As a byproduct, we obtain an Ostrand-type characterization of asymptotic dimension for coarse structures. It is shown that properties like finite asymptotic dimension, coarse finitism, large scale weak paracompactness, ect. are all invari…
The paper studies properties of group relations induced by compatible coarse structures.
problem Properties of asymptotic resemblance relations on groups.
method Generalization of asymptotic dimension and introduction of set theoretic coupling.
result Groups with compatible coarse structures that admit a set theoretic coupling are asymptotic equivalent.
In this paper we compute the discrete fundamental groups of warped cones. As an immediate consequence, this allows us to show that there exist coarsely simply-connected expanders and superexpanders. This also provides a strong coarse invariant of warped cones and implies that many warped cones cannot be coarsely equiva…
The study explores ends in coarse homotopy of proper geodesic spaces.
problem Understanding ends in coarse homotopy of proper geodesic spaces.
method Recontextualizing ends as a functor and proving properties of coarse path components.
result Existence of a natural surjection from coarse path components to ends, not always an injection.
Introduces bounded scale measure and generalizes property A.
problem Defining property A for large scale spaces with bounded geometry.
method Introduces bounded scale measure, shows its coarse invariance, and generalizes property A.
result Definition of property A for large scale spaces with bounded scale measure is a coarse invariant.
Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.
problem Establishing connections between coarse homotopy theory and shape theory.
method Using pointed shape invariants and inverse mapping telescopes.
result Proving two compact spaces are strong shape equivalent if their Euclidean cones are coarsely homotopy equivalent.
Generalizes Bestvina's -boundaries to coarse -boundaries.
problem Establishing properties of -boundaries for groups.
method Introducing a new concept of a 'coarse -boundary' and proving theorems about it.
result Admitting a coarse -boundary is a pure quasi-isometry invariant.
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.
Study on stable mixed commutator length in coarse group theory.
problem Understanding the large scale behavior of stable mixed commutator length in group theory.
method Introducing a bi-invariant metric function and connecting it to coarse group theoretic structures and invariant quasimorphisms.
result Proved that the coarse kernel of the coarse homomorphism is isomorphic to Z^ℓ as a coarse group.
New coarse LS-category introduced for groups and spaces.
problem Large-scale topological properties of groups and spaces.
method Introducing a coarse analog of Lusternik-Schnirelmann category for metric spaces.
result Established lower and upper bounds for geometrically finite and bicombable groups.
We classify up to coarse equivalence all countable abelian groups of finite torsion free rank. The Q-cohomological dimension and the torsion free rank are the two invariants that give us such classification. We also prove that any countable abelian group of finite torsion free rank is coarsely equivalent to Z^n + H whe…
The paper develops glueing theory for topological spaces and applies it to compactifications.
problem Developing a theory for gluing topological spaces and its applications.
method Developed the theory of Artin-Wraith glueings for topological spaces and applied it to compactifications.
result The space of ends of coarse equivalent metric spaces are the same.
Paper proves a relative version of coarse Alexander duality and applies it to Jordan cycles.
problem Proving a relative version of coarse Alexander duality.
method Introduced a relative Čech homology satisfying Eilenberg-Steenrod Exactness Axiom.
result Jordan cycle invariant in proving existence of certain 3-manifold groups.
Atomistic or ab-initio molecular dynamics simulations are widely used to predict thermodynamics and kinetics and relate them to molecular structure. A common approach to go beyond the time- and length-scales accessible with such computationally expensive simulations is the definition of coarse-grained molecular models.…
We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincaré profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini--Schramm--Timár. In this paper we focus on properties of the Poincaré profiles of groups with …
We provide an interpretation of the APS index theorem of Piazza-Schick and Zeidler in terms of coarse homotopy theory. On the one hand we propose a motivic version of the boundary value problem, the index theorem, and the associated secondary invariants. On the other hand, we discuss in detail how the abstract version …
A compact Polish foliated space is considered. Part of this work studies coarsely quasi-isometric invariants of leaves in some residual saturated subset when the foliated space is transitive. In fact, we also use "equi-" versions of this kind of invariants, which means that the definition is satisfied with the same con…
We introduce a geometric property complementary-finite asymptotic dimension (coas- dim). Similar with asymptotic dimension, we prove the corresponding coarse invariant theorem, union theorem and Hurewicz-type theorem.
New theorem shows metrics of certain groups are close if their lengths are identical.
problem Identifying metrics of relatively hyperbolic groups from their lengths.
method Proved rigidity for relatively hyperbolic groups using coarse marked length spectrum.
result Metrics of relatively hyperbolic groups are uniformly close if lengths are identical.
We prove that two countable locally finite-by-abelian groups G,H endowed with proper left-invariant metrics are coarsely equivalent if and only if their asymptotic dimensions coincide and the groups are either both finitely-generated or both are infinitely generated. On the other hand, we show that each countable group…
Study compares metrics from negative curvature and quasi-Fuchsian representations.
problem Comparing metrics on surface groups from negative curvature and quasi-Fuchsian representations.
method Examines Teichmüller space as the intersection of two metric families.
result Teichmüller space is the only common part of the two metric families.
The paper studies properties of optimal metrics associated to curves on surfaces.
problem Investigating properties of optimal metrics associated to curves on surfaces.
method Starting from a filling curve and a separating curve, constructing a two integer parameter family of curves and deriving coarse length bounds and qualitative properties of their associated optimal metrics.
result There are infinitely many pairs of filling curves with distinct inf invariants but the same self-intersection number.
For G a complex reductive group and X a smooth projective or convex quasi-projective polarized G-variety we construct a formal map in quantum K-theory from the equivariant quantum K-theory to the quantum K-theory of the git quotient assuming the quotient is a smooth Deligne-Mumford stack wit…
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.
Coarse free productsmath.GT
We define a notion of free product for coarse spaces that generalizes the corresponding notion of a free product for groups. We show that free products preserve coarse properties such as coarse property C, finite coarse decomposition complexity, and coarse property A. We also give an upper bound estimate on the dimensi…
Defines coarse cohomology of space complements, proving new duality results.
problem Defining and studying coarse cohomology of space complements.
method Introducing a model space, new approach to PD spaces, homological criterion.
result Proves new versions of coarse Poincaré duality and Alexander duality.
Coarse assembly mapsmath.KT
For every strong coarse homology theory we construct a coarse assembly map as a natural transformation between coarse homology theories. We provide various conditions implying that this assembly map is an equivalence. These results generalize known results for the analytic coarse assembly map for K-homology to general …
This paper is a survey of some of the developments in coarse extrinsic geometry since its inception in the work of Gromov. Distortion, as measured by comparing the diameter of balls relative to different metrics, can be regarded as one of the simplist extrinsic notions. Results and examples concerning distorted subgrou…
In this article, we introduce the notion of a functor on coarse spaces being coarsely excisive- a coarse analogue of the notion of a functor on topological spaces being excisive. Further, taking cones, a coarsely excisive functor yields a topologically excisive functor, and for coarse topological spaces there is an ass…
We introduce large scale analogues of topological monotone and light maps, which we call coarsely monotone and coarsely light maps respectively. We show that these two classes of maps constitute a factorization system on the coarse category. We also show how coarsely monotone maps arise from a reflection in a similar w…
Interprets coarse symbol and index classes for Callias type operators.
problem Understanding coarse geometry and index classes for Callias type operators.
method Interprets coarse symbol and index classes in terms of K-theory classes of coarse corona.
result Local positivity and invertibility conditions are incorporated into support conditions in K-theory.
The coarse category was established by Roe to distill the salient features of the large-scale approach to metric spaces and groups that was started by Gromov. In this paper, we use the language of coarse spaces to define coarse versions of asymptotic property C and decomposition complexity. We prove that coarse propert…
We show that coarse property C is preserved by finite coarse direct products. We also show that the coarse analog of Dydak's countable asymptotic dimension is equivalent to the coarse version of straight finite decomposition complexity and is therefore preserved by direct products.
Coarse Homotopy Groupsmath.GT
In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse homotopy groups. The main result is that the coarse homotopy groups of cone of a compa…
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…
New concept of coarse medians for higher rank symmetric spaces.
problem Understanding medians in higher rank symmetric spaces.
method Introducing coarse -median spaces and proving their existence.
result Existence of coarse higher medians on divisible and quasi-homogeneous convex domains.
We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov's notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum-Connes conjecture holds for metric …
Following Roe and others (see, e.g., [MR1451755]), we (re)develop coarse geometry from the foundations, taking a categorical point of view. In this paper, we concentrate on the discrete case in which topology plays no role. Our theory is particularly suited to the development of the_Roe (C*-)algebras_ C*(X) and their K…
Classifies homeomorphism groups of countable Stone spaces up to coarse equivalence.
problem Classifying non-locally compact topological groups using geometric group theory.
method Classification based on coarsely bounded sets and quasi-isometry.
result Groups in the second class are quasi-isometric to the Hamming cube.
Coarse homology of leavesmath.GT
We investigate the coarse homology of leaves in foliations of compact manifolds. This is motivated by the observation that the non-leaves constructed by Schweitzer and by Zeghib all have non-finitely generated coarse homology. This led us to ask whether the coarse homology of leaves in a compact manifold always has to …
Paper relates asymptotic dimension to cofinal dimension using coarse proximities.
problem Relating asymptotic dimension to cofinal dimension in metric spaces.
method Introducing coarse proximities and inverse limit constructions.
result Asymptotic dimension is bounded by coarse cofinal dimension and cofinal dimension of Higson corona.
Unique median structures found in hyperbolic spaces.
problem Uniqueness of median structures in hyperbolic spaces.
method Analyzing product of hyperbolic spaces and properties of relative hyperbolicity.
result Non-hyperbolic pants graphs can have unique median structures.
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.
The paper shows how coarse embeddings affect homological Dehn functions.
problem Characterizing groups with coarse embeddings into hyperbolic groups.
method Demonstrates a coarse embedding theorem for homological filling functions.
result Characterizes groups with coarse embeddings into hyperbolic groups of geometric dimension 2.
Lifting coarse homotopiesmath.GT
Coarse geometry, and in particular coarse homotopy theory, has proven to be a powerful tool for approaching problems in geometric group theory and higher index theory. In this paper, we continue to develop theory in this area by proving a Coarse Lifting Lemma with respect to a certain class of bornologous surjective ma…