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 CAT(0) metrics are arborescent.
problem Characterizing locally finite complexes with CAT(0) metrics. method Proving arborescence for complexes with polyhedral CAT(0) metrics. result Locally finite complexes with polyhedral CAT(0) metrics are arborescent. 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.
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 exp(n)(xm) and exp(n)(xα). 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).
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 Γ is said to be {\it rigid}, if Γ 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.
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.
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.
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 Γ 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.
Lecture notes on CAT(0) cube complexes for beginners.
problem Introduction to CAT(0) cube complexes for students.
method Elementary and self-contained lectures.
result Accessible introduction to advanced topics for students.
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.
New examples show right-angled Artin groups can have connected boundaries.
problem CAT(0) group boundaries not always path connected.
method Provided examples of right-angled Artin groups with connected boundaries.
result Right-angled Artin groups can have boundaries that are path connected.
Path connectedness of boundaries for certain CAT(0) groups with isolated flats.
problem Conditions for the path connectedness of boundaries of CAT(0) groups.
method Study of CAT(0) groups with isolated flats acting on CAT(0) spaces.
result Visual boundaries of CAT(0) groups with isolated flats are path connected.
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.
A simpler proof shows L2-metric completion is CAT(0).
problem Completing Riemannian metrics space.
method Easier proof of existing result by Brian Clarke.
result Completion of Riemannian metrics is CAT(0).
CAT is a new ASR toolkit using CRF and CTC for state-of-the-art speech recognition.
problem Improving automatic speech recognition systems.
method CRF-based discriminative training with CTC-inspired state topology.
result CAT achieves state-of-the-art results with fewer parameters and is competitive with hybrid models.
In this paper, we study the capacity dimension of the boundary of CAT(0) spaces. We first compare the two metrics on the boundary of a hyperbolic CAT(0) space, i.e., the visual metric and the conical metric, and prove that they give the same capacity dimension of the boundary. Then we study the capacity dimension o…
Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is CAT(0) with a metric for which all vertex stars are convex. This strengthens and generalizes a result by Crowley. Further consequences of our work are: (1) …
CAT toolkit combines hybrid and E2E approaches for efficient speech recognition.
problem Improving speech recognition efficiency and latency.
method CTC-CRF based framework with contextualized soft forgetting.
result CAT achieves state-of-the-art results with simpler training and streaming ASR.
We prove that every finite-volume hyperbolic 3-manifold M with p > 0 cusps admits a canonical, complete, piecewise Euclidean CAT(0) metric, with a canonical projection to a CAT(0) spine K. Moreover, (a) the universal cover of M endowed with the CAT(0) metric is a union of Euclidean half-spaces, glued together by identi…
Study on infinite energy maps from surfaces to CAT(0) spaces.
problem Harmonic maps with infinite energy from Riemann surfaces to CAT(0) spaces.
method Estimates of energy growth near punctures, proof of uniqueness.
result Precise estimates of energy growth near punctures and proof of uniqueness of harmonic maps.
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.
CAT(0) spaces close to Euclidean spheres are homeomorphic to Euclidean spaces.
problem Understanding geometric properties of CAT(0) spaces near spheres.
method Analyzing the relationship between CAT(0) spaces and their Tits boundaries.
result CAT(0) spaces close to Euclidean spheres are homeomorphic to Euclidean spaces.
The study glues CAT(0) subsets of the plane under certain curvature conditions.
problem Understanding how to combine locally CAT(0) spaces. method Describes a gluing process for subsets of the Euclidean plane that maintain the CAT(0) property. result The gluing of two CAT(0) subsets, under specific curvature conditions, results in another CAT(0) space. We study affine maps between CAT(0) spaces with geometric actions, and show that they essentially split as products of dilations and linear maps (on the Euclidean factor). This extends known results from the Riemannian case. Furthermore, we prove a splitting lemma for the Tits boundary of a CAT(0) space with geometric …
We study discrete groups from the view point of a dimension gap in connection to CAT(0) geometry. Developing studies by Brady-Crisp and Bridson, we show that there exist finitely presented groups of geometric dimension 2 which do not act properly on any proper CAT(0) spaces of dimension 2 by isometries, although such a…
Let $\imath: M\to \RR^{p+2}$ be a smooth embedding from a connected, oriented, closed p-dimesional smooth manifold to $\RR^{p+2}$, then there is a spin structure ♯(ςp+2) on M canonically induced from the embedding. If an orientation-preserving diffeomorphism τ of M extends over as an o…
We construct CAT(0) groups containing subgroups whose Dehn functions are given by xs, for a dense set of numbers s∈[2,∞). This significantly expands the known geometric behavior of subgroups of CAT(0) groups.
New method connects CAT(0) spaces to hyperbolic spaces.
problem Injecting sublinear Morse boundaries into Gromov boundaries.
method Developed curtain machinery to characterize sublinear Morse properties.
result Continuous injection of sublinear Morse boundaries into Gromov boundaries.
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.
Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.
problem CAT(0) spaces of higher rank with geometric group actions.
method Proving rigidity for spaces containing periodic flats and geodesics in flats.
result CAT(0) spaces of higher rank n≥2 are rigid if they contain a periodic n-flat. CAT(0) properties extended to a class of Shephard groups.
problem Extending CAT(0) properties to Shephard groups.
method Introduced new criteria for CAT(1) complexes and extended results for Coxeter groups to Shephard groups.
result Shephard groups with enough finite parabolic subgroups are CAT(0).
The paper presents counterexamples to LS-category conjectures and constructs maps between manifolds.
problem Counterexamples to LS-category conjectures for manifolds and their squares.
method Construction of manifolds and maps to analyze LS-category properties.
result Shows that mcatLS(M2imesM3)≥4 and reduces Rudyak's conjecture. Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.
problem Proving Lipschitz regularity for harmonic map heat flows into CAT(0) spaces.
method Elliptic approximation method
result Every weak solution of the harmonic map heat flow into CAT(0) spaces is Lipschitz continuous in both space and time.
We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension d for all d≥4. These examples result from applying CAT(0)/CAT(−1) filling constructions (based on singular doubly warped products) to finite volume hyperbolic manifolds with toral cusps. The groups obtained have a…
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…
Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
problem Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
method Proving non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
result Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
New group acts on complex but not in lower dimensions.
problem Understanding group actions on cube complexes.
method Constructed a specific group acting on a locally finite CAT(0) cube complex.
result Group does not act properly on a finite dimensional CAT(0) cube complex.
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}.