In this paper, we study CAT(0) spaces with non-locally connected boundary. We give some condition of a CAT(0) space whose boundary is not locally connected.
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Deep learning CAT bond valuationq-fin.PR
Deep learning speeds CAT bond valuation.
problem Valuation of Catastrophe bonds.
method Deep neural networks trained to price CAT bonds.
result Trained model provides fast and accurate pricing.
The study finds conditions for groups acting on CAT(0) cube complexes to have infinite girth.
problem Conditions for groups acting on CAT(0) cube complexes to have infinite girth.
method Analyzes lattices in automorphism groups of finite dimensional CAT(0) cube complexes.
result Groups either have infinite girth or are {locally finite}-by-{virtually abelian}.
On CAT() surfacesmath.MG
This paper proves properties of CAT(κ) surfaces with bounded curvature.
problem Properties of CAT(κ) surfaces with bounded curvature.
method Proof of bounded curvature, approximation by smooth surfaces.
result CAT(κ) surfaces have bounded curvature.
Gluing domainsmath.DG
The study glues subsets of the plane under certain curvature conditions.
problem Understanding how to combine locally spaces.
method Describes a gluing process for subsets of the Euclidean plane that maintain the property.
result The gluing of two subsets, under specific curvature conditions, results in another space.
We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the CAT(k) inequality. We prove a general CAT(k) extension theorem, giving sufficient conditions on and near the boundary of a locall…
In this paper, we investigate an equivariant homeomorphism of the boundaries and of two proper CAT(0) spaces and on which a CAT(0) group acts geometrically. We provide a sufficient condition and an equivalent condition to obtain a -equivariant homeomorphism of the boundaries $\p…
Busemann G-spaces with Finsler metrics
problem Admitting a differentiable DC atlas with a continuous Finsler metric
method Comparison geometry
result Generalizes previous work on G-spaces with Riemannian curvature bounds
In this paper, we present a new open source toolkit for automatic speech recognition (ASR), named CAT (CRF-based ASR Toolkit). A key feature of CAT is discriminative training in the framework of conditional random field (CRF), particularly with connectionist temporal classification (CTC) inspired state topology. CAT co…
New conditions ensure surface group extensions are non-positively curved.
problem Conditions for surface group extensions to be CAT(0).
method Generalized necessary conditions for surface-by-surface groups.
result If is CAT(0) with infinite monodromy, the monodromy representation has a finite kernel.
In this article, I prove that full subcomplexes of CAT(0) simplicial 3-complexes inherit the non positive curvature condition, and describe a family of counterexamples that prove this result can not be extended to higher dimensions.
We introduce new techniques for studying boundary dynamics of CAT(0) groups. For a group acting geometrically on a CAT(0) space we show there is a flat of maximal dimension whose boundary sphere intersects every minimal -invariant subset of . As a result we derive a necessary …
The study proves conditions for CAT(0) spaces with higher rank rigidity.
problem Conditions for rigidity in CAT(0) spaces with higher rank.
method Geometric group action and analysis of geodesics and flats.
result CAT(0) spaces with specific properties are classified.
In this paper, we investigate an equivariant homeomorphism of the boundaries and of two proper CAT(0) spaces and on which a CAT(0) group acts geometrically. We provide a sufficient condition to obtain a -equivariant homeomorphism of the two boundaries and $\partial …
We prove that a hyperplane in a CAT(0) cubical complex X has no self-intersections and separates X into two convex complementary components. These facts were originally proved by Sageev. Our argument shows that his theorem is a corollary of Gromov's link condition. We also give new arguments establishing some combinato…
We investigate the Tits boundary of locally compact CAT(0) 2-complexes. In particular we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary t…
For a k-flat F inside a locally compact CAT(0)-space X, we identify various conditions that ensure that F bounds a (k+1)-dimensional half flat in X. Our conditions are formulated in terms of the ultralimit of X. As applications, we obtain (1) constraints on the behavior of quasi-isometries between tocally compact CAT(0…
We construct examples of smooth 4-dimensional manifolds M supporting a locally CAT(0)-metric, whose universal cover X satisfy Hruska's isolated flats condition, and contain 2-dimensional flats F with the property that the boundary at infinity of F defines a nontrivial knot in the boundary at infinity of X. As a consequ…
It is well known that every word hyperbolic group has a well-defined visual boundary. An example of C. Croke and B. Kleiner shows that the same cannot be said for CAT(0) groups. All boundaries of a CAT(0) group are, however, shape equivalent, as observed by M. Bestvina and R. Geoghegan. Bestvina has asked if they also …
The paper defines a new metric space invariant and computes it for various manifolds.
problem Computing the distributional LS-category of manifolds.
method Defining and analyzing the distributional LS-category of metric spaces and applying it to manifolds.
result Several sufficient conditions for the distributional LS-category of a closed manifold to be maximum are derived.
Automorphisms of contact graphs match those of cube complexes under weak conditions.
problem Understanding automorphisms of cube complexes and their contact graphs.
method Analyzing the relationship between automorphisms of cube complexes and their contact graphs.
result The automorphism groups of cube complexes and their contact graphs coincide under weak assumptions.
The study establishes conditions for groups acting on polygonal complexes to contain virtually free subgroups.
problem Conditions for groups acting on polygonal complexes to contain virtually free subgroups.
method Analysis of links of polygonal complexes and conditions on their structure.
result Groups acting on polygonal complexes with certain link conditions contain virtually free subgroups.
Locally connected boundaries of Coxeter groups are studied.
problem Conditions for locally connected boundaries of Coxeter groups.
method Analyzes the defining graph of right-angled Coxeter groups.
result Guarantees locally connected boundaries of CAT(0) spaces.
Let G be a simple algebraic group. Labelled trivalent graphs called webs can be used to product invariants in tensor products of minuscule representations. For each web, we construct a configuration space of points in the affine Grassmannian. Via the geometric Satake correspondence, we relate these configuration spaces…
Study shows Poisson boundary matches hyperbolic boundary for certain groups.
problem Identifying Poisson boundary for hyperbolic groups without moment conditions.
method Proved using finite entropy random walks and extended to groups with WPD elements.
result Poisson boundary matches hyperbolic boundary for specified groups.
We develop an algorithm of polynomial time complexity to construct the Grushko decomposition of fundamental groups of graphs of free groups with cyclic edge groups. Our methods rely on analysing vertex links of certain CAT(0) square complexes naturally associated with a special class of the above groups. Our main resul…
Study of geometric actions on CAT(0) spaces and their limits.
problem Understanding limits of geometric actions on CAT(0) spaces.
method Analysis of convergence and splitting/collapsing phenomena in CAT(0) lattices and orbispaces.
result Proof of compactness theorem for CAT(0) homology orbifolds.
CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
problem Characterizing CAT(0) spaces with specific volume growth properties.
method Analyzing asymptotic topological regularity and volume growth.
result CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
Locally finite complexes with polyhedral metrics are arborescent.
problem Characterizing locally finite complexes with metrics.
method Proving arborescence for complexes with polyhedral metrics.
result Locally finite complexes with polyhedral metrics are arborescent.
M. Bestvina has shown that for any given torsion-free CAT(0) group G, all of its boundaries are shape equivalent. He then posed the question of whether they satisfy the stronger condition of being cell-like equivalent. In this article we prove that the answer is "Yes" in the situation where the group in question splits…
New CAT(0) groups show superexponential subgroup Dehn functions.
problem Understanding subgroups of CAT(0) groups with superexponential Dehn functions.
method Construction of specific 4- and 6-dimensional CAT(0) groups.
result Found subgroups with Dehn functions and .
CAT(0) study of curve complements and families.
problem Understanding CAT(0) behavior of curve complements and families.
method Investigation of fundamental groups of plane curve complements and universal families.
result Fundamental groups of certain families are not CAT(0).
Braid group proof shows 7-strand is CAT(0).
problem Proving the 7-strand braid group is CAT(0).
method Elaborating on Haettel, Kielak, and Schwer's argument.
result The 7-strand braid group is proven to be CAT(0).
On splitting theorems for CAT(0) spaces and compact geodesic spaces of non-positive curvaturemath.GR
In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group is said to be {\it rigid}, if determines the boundary up to homeomorphisms of a CAT(0) space on whi…
Asymptotically CAT(0) metrics and Z-structures for HHGs, proving Farrell-Jones Conjecture.
problem Proving Farrell-Jones Conjecture for HHGs.
method Asymptotically CAT(0) metrics and Z-structures construction.
result Many HHGs satisfy Farrell-Jones Conjecture.
Toyoda proved 5-point CAT(0) spaces; this paper offers a new proof.
problem Describing 5-point subspaces in CAT(0) length spaces.
method Provides an alternative proof of Toyoda's theorem.
result Offers a new proof of Toyoda's theorem.
We show that groups satisfying Kazhdan's property (T) have no unbounded actions on finite dimensional CAT(0) cube complexes, and deduce that there is a locally CAT(-1) Riemannian manifold which is not homotopy equivalent to any finite dimensional, locally CAT(0) cube complex.
We discuss a problem posed by Gersten: Is every automatic group which does not contain Z+Z subgroup, hyperbolic? To study this question, we define the notion of "n-tracks of length n", which is a structure like Z+Z, and prove its existence in the non-hyperbolic automatic groups with mild conditions. As an application, …
We generalize the theory of gradient flows of semi-convex functions on CAT(0)-spaces, developed by Mayer and Ambrosio--Gigli--Savaré, to CAT(1)-spaces. The key tool is the so-called "commutativity" representing a Riemannian nature of the space, and all results hold true also for metric spaces satisfying the commutativi…
Affine cactus groups are CAT(0) and hyperbolic.
problem Characterizing geometric properties of affine cactus groups.
method Analyzing CAT(0) and hyperbolic properties through group theory.
result Affine cactus groups of degree three are hyperbolic.
Study mapping class groups on CAT(0) cube complexes.
problem Asymptotically rigid mapping class groups of infinitely-punctured surfaces.
method Introduced CAT(0) cube complexes and determined their properties.
result Determined CAT(0) properties of cube complexes.
In this paper, we investigate finitely generated groups of isometries of CAT(0) spaces containing some central hyperbolic isometry, and study CAT(0) groups. We show that every CAT(0) group has the semi-direct product structure $Γ=(\cdots(((Γ'\rtimes\langleδ_{n}\rangle)\rtimes\langleδ_{n-1}\rangle)\rtimes\langleδ_{n…
Groups on CAT(0) cube complexes grow exponentially uniformly.
problem Uniform exponential growth of groups acting on CAT(0) cube complexes.
method Study groups acting without global fixed points on CAT(0) square complexes.
result Groups with uniform exponential growth or stabilize Euclidean subcomplexes.
We extend the methods of Davis-Januszkiewicz-Lafont to provide a new obstruction to smooth Riemannian metric with non-positive sectional curvature. We construct examples of locally CAT(0) 4-manifolds , whose universal covers satisfy isolated flats condition and contain 2-dimensional flats with the property that $\sq…
Geodesics spiral around compact subsets in CAT(0) spaces.
problem Understanding geodesic spiraling in CAT(0) spaces.
method Logarithm law-type result for geodesics in quotients of rank one CAT(0) spaces.
result Proved logarithm law for geodesic spiraling in certain CAT(0) spaces.
CAT(0) spaces quasi-isometric to Euclidean spaces are homeomorphic to R^n.
problem Characterizing CAT(0) spaces quasi-isometric to Euclidean spaces.
method Analyzing properties of CAT(0) spaces and using quasi-isometry.
result CAT(0) spaces quasi-isometric to R^n are not necessarily homeomorphic to it.
We examine volume pinching problems of CAT(1) spaces. We characterize a class of compact geodesically complete CAT(1) spaces of small specific volume. We prove a sphere theorem for compact CAT(1) homology manifolds of small volume. We also formulate a criterion of manifold recognition for homology manifolds on volume g…
We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.