arXiv research

Asymptotically CAT(0) metrics and Z-structures for HHGs, proving Farrell-Jones Conjecture.

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 $\imath^\sharp(ς^{p+2})$ on $M$ canonically induced from the embedding. If an orientation-preserving diffeomorphism $τ$ of $M$ extends over $\imath$ as an o…

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…

Study on median algebra structures on Euclidean spaces and manifolds with local CAT(0) cubulation.

We announce results on the structure of CAT(0) groups, CAT(0) lattices and of the underlying spaces. Our statements rely notably on a general study of the full isometry groups of proper CAT(0) spaces. Classical statements about Hadamard manifolds are established for singular spaces; new arithmeticity and rigidity state…

We prove that a minimal disc in a CAT(0) space is a local embedding away from a finite set of "branch points". On the way we establish several basic properties of minimal surfaces: monotonicity of area densities, density bounds, limit theorems and the existence of tangent maps. As an application, we prove Fary-Milnor's…

New findings on geometric flows and equidistribution in Hilbert geometry.

New Coxeter groups have unique boundary structures.

CATNet predicts CAT bond spreads using graph-based deep learning.

Mahan Mitra (Mj) proved Cannon--Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group. We prove that Cannon--Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic CAT(0) groups with isolated flats with respect to the visual boundaries. We also show Cannon--Thurston ma…

CAT method learns causal structure of directed trees efficiently.

Homotopy equivalent boundaries of cube complexes are studied.

Paper uses machine learning to uncover nonlinear dynamics in CAT bond pricing.

We study complex surfaces with locally CAT(0) polyhedral Kahler metrics and construct such metrics on CP^2 with various orbifold structures. In particular, in relation to questions of Gromov and Davis-Moussong we construct such metrics on a compact quotient of the two-dimensional unite complex ball. In the course of th…

CaTs use DAGs with transformers to enforce causal constraints, improving neural network robustness.

We prove that every open subset of a euclidean building is a finite dimensional absolute neighborhood retract. This implies in particular that such a set has the homotopy type of a finite dimensional simplicial complex. We also include a proof for the rigidity of homeomorphisms of euclidean buildings. A key step in our…

We classify Dehn functions of Bestvina-Brady groups.

New method proves heat flow of harmonic maps into CAT(0) spaces.

Busemann G-spaces with Finsler metrics

Study of geometric actions on CAT(0) spaces and their limits.

CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.

Locally finite complexes with polyhedral $CAT(0)$ metrics are arborescent.

This paper proves properties of CAT(κ) surfaces with bounded curvature.

New CAT(0) groups show superexponential subgroup Dehn functions.

CAT(0) study of curve complements and families.

CaT-GNN improves credit card fraud detection by integrating causal reasoning into GNNs.

Suppose a group $G$ is relatively hyperbolic with respect to a collection $\PP$ of its subgroups and also acts properly, cocompactly on a $\CAT(0)$ (or $δ$--hyperbolic) space $X$. The relatively hyperbolic structure provides a relative boundary $\partial(G,\PP)$. The $\CAT(0)$ structure provides a different boundary at…

Braid group proof shows 7-strand is 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…

Classifies 3-manifold groups with equivariant hierarchically hyperbolic structures.

Toyoda proved 5-point CAT(0) spaces; this paper offers a new proof.

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 give a generalized and self-contained account of Haglund-Paulin's wallspaces and Sageev's construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to pro…

The paper extends geometric results from negatively-curved spaces to strictly convex Hilbert geometry.

This work achieves exponential concentration in heavy-tailed data over CAT(κ) spaces using the Fréchet median.

Let $X$ be a proper, geodesically complete CAT(0) space under a proper, non-elementary, isometric action by a group $Γ$ with a rank one element. We construct a generalized Bowen-Margulis measure on the space of unit-speed parametrized geodesics of $X$ modulo the $Γ$-action. Although the construction of Bowen-Margulis m…

Affine cactus groups are CAT(0) and hyperbolic.

Study mapping class groups on CAT(0) cube complexes.

Groups on CAT(0) cube complexes grow exponentially uniformly.

Geodesics spiral around compact subsets in CAT(0) spaces.

CAT(0) spaces quasi-isometric to Euclidean spaces are homeomorphic to R^n.

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.

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 $L^2$-metric completion is CAT(0).

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.