arXiv research

The paper compactifies spaces of half-translation structures on surfaces.

Characterizes fundamental groups of disjointly tree-graded spaces.

We define and give explicit construction of the universal tree-graded space with a given collection of pieces. We apply that to proving uniqueness of asymptotic cones of relatively hyperbolic groups whose peripheral subgroups have unique asymptotic cones. Modulo the Continuum Hypothesis, we show that if an asymptotic c…

We study the bilipschitz equivalence type of tree-graded spaces, showing that asymptotic cones of relatively hyperbolic groups (resp. asymptotic cones of groups containing a cut-point) only depend on the bilipschitz equivalence types of the pieces in the standard (resp. minimal) tree-graded structure. In particular, th…

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asy…

We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic g…

Given a metric space $X$ of finite asymptotic dimension, we consider a quasi-isometric invariant of the space called dimension function. The space is said to have asymptotic Assouad-Nagata dimension less or equal $n$ if there is a linear dimension function in this dimension. We prove that if $X$ is a tree-graded space …

We prove the equivalence between a relative bottleneck property and being quasi-isometric to a tree-graded space. As a consequence, we deduce that the quasi-trees of spaces defined axiomatically by Bestvina-Bromberg-Fujiwara are quasi-isometric to tree-graded spaces. Using this we prove that mapping class groups quasi-…

Study compactifies representations space of hyperbolic surfaces.

In this work, we study the asymptotic geometry of the mapping class group and Teichmueller space. We introduce tools for analyzing the geometry of `projection' maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. We deduce several a…

CAT(0) spaces can be deformed in various ways.

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

Metric minimizing surfaces are locally CAT(0) in CAT(0) spaces.

Study pinches volume of CAT(1) spaces, proving sphere theorem and manifold recognition criterion.

The paper examines the capacity dimension of boundaries in CAT(0) spaces.

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…

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

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

CAT(-1) properties shown for Teichmüller geodesics in thick moduli space.

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

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

A simpler proof shows $L^2$-metric completion is CAT(0).

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.

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

Groups with certain actions on CAT(0) spaces have global fixed points.

Study on infinite energy maps from surfaces to CAT(0) spaces.

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

Minimal surfaces in CAT(0) spaces are well-behaved except at a few points.

Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.

The study proves Liouville-type theorems and Bochner formulas for harmonic maps into CAT(κ) spaces.

Maximal open subset found for product of CAT(-1) spaces.

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 …

Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.

Study shows how maps from certain geometric spaces behave near their edges.

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

New coarse LS-category introduced for groups and spaces.

Groups acting on CAT(0) spaces without 3-flats have rigid properties.

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

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…

Study on harmonic maps in special geometric spaces.

Under certain assumptions on CAT(0) spaces, we show that the geodesic flow is topologically mixing. In particular, the Bowen-Margulis' measure finiteness assumption used in recent work of Ricks is removed. We also construct examples of CAT(0) spaces which do not admit finite Bowen-Margulis measure.

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…

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…

The paper proves local Lipschitz regularity for harmonic map heat flows in RCD spaces into CAT(0) spaces.

New examples show right-angled Artin groups can have connected boundaries.

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…

We construct several non-trivial examples of CAT(1) spaces by using the idea of free construction.