arXiv research

We consider a cocompact discrete reflection group $W$ of a CAT(0) space $X$. Then $W$ becomes a Coxeter group. In this paper, we study an analogy between the Davis-Moussong complex $Σ(W,S)$ and the CAT(0) space $X$, and show several analogous results about the limit set of a parabolic subgroup of the Coxeter group $W$.

We show that the translation length of any parabolic isometry on a complete semi-uniformly visible CAT(0) space is always zero. As a consequence, we will classify the isometries on visible CAT(0) spaces in terms of translation lengths. We will also show that the moduli space $\mathbb{M}(S_{g,n})$ of surface $S_{g,n}$ o…

Strong bolicity helps prove Baum-Connes conjecture for certain hyperbolic groups.

We study the fixed point set in the ideal boundary of a parabolic isometry of a proper CAT(0)-space. We show that the radius of the fixed point set is at most pi/2, and study its centers. As a consequence, we prove that the set of fixed points is contractible with respect to the Tits topology.

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

CAT(0) properties extended to a class of Shephard groups.

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…

Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.

The study restricts matrix group actions on CAT(0) spaces and uniquely arcwise connected spaces, proving fixed points are inevitable.

Clarifies boundary criterion for non-one-ended subgroups in cubulation theory.

Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give a complete topological characterization of the Martin boundary of finitely supported random walks on relatively hyperbolic groups with virt…

We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residua…

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.

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…

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

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.

Affine cactus groups are CAT(0) and hyperbolic.

Study mapping class groups on CAT(0) 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 $Γ$ 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.

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.

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…

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.

CAT(0) spaces close to Euclidean spheres are homeomorphic to Euclidean spaces.

In all known examples of a CAT(0) group acting on CAT(0) spaces with non-homeomorphic CAT(0) visual boundaries, the boundaries are each not path connected. In this paper, we show this does not have to be the case by providing examples of right-angled Artin groups which exhibit non-unique CAT(0) boundaries where all of …

Deep learning speeds CAT bond valuation.

The study glues $CAT(0)$ subsets of the plane under certain curvature conditions.

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 …

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…

We construct CAT(0) groups containing subgroups whose Dehn functions are given by $x^s$, for a dense set of numbers $s \in [2, \infty)$. This significantly expands the known geometric behavior of subgroups of CAT(0) groups.

New method connects CAT(0) spaces to hyperbolic spaces.

New coarse LS-category introduced for groups and spaces.