arXiv research

CAT(0) study of curve complements and families.

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

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 findings on geometric flows and equidistribution in Hilbert geometry.

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) …

These notes grew out of two lectures I have given on CAT(0) cube complexes. I've tried to keep the material elementary and self-contained in order to keep the material easily accessible and to provide an elementary introduction on the topic for advanced bachelor students or early master students with little to no previ…

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

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…

The study shows that certain complex geometries are hyperbolic and contractible but fail to be CAT(0).

Researchers solve 3D cube complex boundary rigidity problem.

We show that the geometry of a Riemannian manifold (M,g) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted cat_{LS}(M). Here we introduce a Riemannian analogue of cat_{LS}(M), called the systolic category of M. It is denoted cat_{sys}(M), and d…

Uniform convexity in divisible domains leads to hyperbolic geometry.

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…

Busemann G-spaces with Finsler metrics

New proof of Kobayashi's properness criterion using metric geometry.

Local Lipschitz continuity of sub-elliptic harmonic maps into CAT(0) spaces proved.

In this paper, we study the geometry of cone-offs of CAT(0) cube complexes over a family of combinatorially convex subcomplexes, with an emphasis on their Gromov-hyperbolicity. A first application gives a direct cubical proof of the characterization of the (strong) relative hyperbolicity of right-angled Coxeter groups,…

Study rigidity of real moment-angle manifolds using cubical geometry.

We present a view of the current understanding of the geometry of Weil-Petersson (WP) geodesics on the completion of the Teichmüller space. We sketch a collection of results by other authors and then proceed to develop the properties of the WP CAT(0) geometry. Our approach includes a simplified proof of the Masur-Wolf …

Develops geometric foundations for sublinear Morse boundaries in mapping class groups and Teichmüller spaces.

New method glues Lorentzian spaces, preserving curvature bounds.

Our goal is to show the beauty and power of Alexandrov geometry by reaching interesting applications and theorems with a minimum of preparation. The topics include 1. Reshetnyak's gluing theorem, 2. Estimates on the number of collisions in billiards, 3. Reshetnyak's majorization theorem, 4. Hadamard--Cartan globalizati…

We investigate the combinatorial and geometric properties of automorphism groups of universal right-angled Coxeter groups, which are the automorphism groups of free products of copies of Z_2. It is currently an open question as to whether or not these automorphism groups have non-positive curvature. Analogous to Outer …

The study improves harmonic map theory for metric spaces with curvature bounds.

This is a survey paper on the topic of Weil-Petersson geometry of Teichmuller spaces. Even though historically the subject has been developed as a branch of complex analysis, the treatment here is from the view-point of differential geometry, much influenced by the works of Eells, Earle, Fischer, Tromba and Wolpert ove…

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

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

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

For a Jordan domain in the plane the length metric space of points connected to an interior point by a curve of finite length is a CAT(0)space and Gromov hyperbolic. With respect to the cone topology, that space plus its boundary at infinity is topologically the same as the original Jordan domain.

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

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.

We develop a notion of rank one properly convex domains (or Hilbert geometries) in the real projective space. This is in the spirit of rank one non-positively curved Riemannian manifolds and CAT(0) spaces. We define rank one isometries for Hilbert geometries and characterize them as being equivalent to contracting elem…

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

Decomposes axis bundles into cubist structures for fully irreducible outer automorphisms.

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.

Survey on harmonic maps in non-smooth spaces, focusing on rigidity.

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).