arXiv research

Minimal polynomial found for Riemannian C_0-spaces.

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

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

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 …

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

The study proves conditions for CAT(0) spaces with higher rank rigidity.

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…

For a k-flat F inside a locally compact CAT(0)-space X, we identify various conditions that ensure that F bounds a (k+1)-dimensional half flat in X. Our conditions are formulated in terms of the ultralimit of X. As applications, we obtain (1) constraints on the behavior of quasi-isometries between tocally compact CAT(0…

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

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

New rigidity result for CAT(0) spaces of higher rank.

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

Computes minimal polynomials for generalized Heisenberg groups.

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

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

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

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.

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…

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

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

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

Constructs CAT(0) actions for certain groups without unipotent elements.

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.

Given a fixed closed manifold M, we exhibit an explicit formula for the distance function of the canonical L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on M. Additionally, we examine the (metric) completion of the manifold of metrics with respect to the L^2 metric and show that there exists a …

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

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

Study shows no new Euclidean factors can appear in the limit of CAT(0) spaces.

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

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 associate to a CAT(0)-space a flow space that can be used as the replacement for the geodesic flow on the sphere tangent bundle of a Riemannian manifold. We use this flow space to prove that CAT(0)-group are transfer reducible over the family of virtually cyclic groups. This result is an important ingredient in our …

The study proves fixed-point theorems for groups acting on CAT(0) spaces.

We give a criterion for group elements to have fixed points with respect to a semi-simple action on a complete CAT(0) space of finite topological dimension. As an application, we show that Thompson's group T and various generalizations of Thompson's group V have global fixed points when they act semi-simply on finite-d…

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…

We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.

The paper studies maps and reducibility for cocycles into CAT(0)-spaces.

We prove that in CAT(0) spaces a quasi-geodesic is Morse if and only if it is contracting. Specifically, in our main theorem we prove that for $γ$ a quasi-geodesic in a CAT(0) space X, the following four statements are equivalent: (i) $γ$ is Morse, (ii) $γ$ is (b,c)--contracting, (iii), $γ$ is strongly contracting, and…

We prove an explicit equivalence between various hyperbolic type properties for quasi-geodesics in CAT(0) spaces. Specifically, we prove that for X a CAT(0) space and $γ$ a quasi-geodesic, the following four statements are equivalent and moreover the quantifiers in the equivalences are explicit: (i) $γ$ is S-Slim, (ii)…

Whenever the mapping class group of a closed orientable surface of genus g acts by semisimple isometries on a complete CAT(0) space of dimension less than g it fixes a point.

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…

New result on group actions in CAT(0) spaces with vanishing escape rate.

In this paper, we investigate a proper CAT(0) space $(X,d)$ which is homeomorphic to $R^2$ and we show that the asymptotic dimension $asdim (X,d)$ is equal to 2.

As demonstrated by Croke and Kleiner, the visual boundary of a CAT(0) group is not well-defined since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. We introduce a new type of boundary for a CAT(0) space, called the contracting boundary, made up rays satisfying one of five hyperbolic-like propertie…

Let S be an orientable surface of finite type and let Mod(S) be its mapping class group. We consider actions of Mod(S) by semisimple isometries on complete CAT(0) spaces. If the genus of S is at least 3, then in any such action all Dehn twists act as elliptic isometries. The action of Mod(S) on the completion of Teichm…

This paper studies convergence of horospheres in CAT(0) spaces.

New criterion for generating free groups in CAT(0) spaces.

Study boundary actions on CAT(0) spaces, proving topological freeness.

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