Cocompact CAT(0) spaces are almost geodesically complete

CAT$(0)$ 空间的 GH 收敛：欧几里得因子的稳定性

GH-convergence of CAT$(0)$-spaces: stability of the Euclidean factor

本文研究带一致余紧离散等距群的测地完备 CAT$(0)$ 空间在 Gromov–Hausdorff 收敛下欧几里得因子的稳定性，证明极限不会产生额外欧几里得因子。

A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries

We prove the following rank rigidity result for proper CAT(0) spaces with one-dimensional Tits boundaries: Let $Γ$ be a group acting properly discontinuously, cocompactly, and by isometries on such a space $X$. If the Tits diameter of $\partial X$ equals $π$ and $Γ$ does not act minimally on $\partial X$, then $\partial X$ is a spherical building or a spherical join. If $X$ is also geodesically complete, then $X$ is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of $\partial X$, doe…

Isometries of spaces of convex compact subsets of CAT(0)-spaces

In the present paper we characterize the surjective isometries of the space of compact, convex subsets of proper, geodesically complete CAT(0)-spaces in which geodesics do not split, endowed with the Hausdorff metric. Moreover, an analogue characterization of the surjective isometries of the space of compact subsets of a proper, uniquely geodesic, geodesically complete metric space in which geodesics do not split, when endowed with the Hausdorff metric, is given.

CAT(0)空间近似欧几里得空间

CAT(0) spaces quasi-isometric to Euclidean spaces

研究显示，满足特定条件的CAT(0)同调流形若近似欧几里得空间则与其同胚，同时存在非同胚的CAT(0)空间近似欧几里得空间。最后提供一个非同调流形序列，证明拓扑流形集在CAT(0)空间类中非开。

余紧凸集边界的可缩性与极限集的嵌入

Contractibility of boundaries of cocompact convex sets and embeddings of limit sets

本文给出CAT(0)-空间中余紧凸集边界分量可缩的充分条件，并应用于负曲率流形群拟凸余维一子群极限集的“野”性研究，证明基于粗上曲率界与插值分析。

曲率有上界的测地完备空间上的非临界映射

Noncritical maps on geodesically complete spaces with curvature bounded above

本文研究了曲率有上界的测地完备空间上距离映射的正则性，证明了其局部为Hurewicz纤维化，并作为推论得到了测地完备CAT(1)空间的球面定理。

Hyperbolic groups with planar boundaries

We prove that the class of convex-cocompact Kleinian groups is quasi-isometrically rigid. We also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually a convex-cocompact Kleinian group provided that its boundary has Ahlfors regular conformal dimension strictly less than $2$ or if it acts geometrically on a CAT(0) cube complex.

A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry

We study geodesically complete and locally compact Hadamard spaces X whose Tits boundary is a connected irreducible spherical building. We show that X is symmetric iff complete geodesics in X do not branch and a Euclidean building otherwise. Furthermore, every boundary equivalence (cone topology homeomorphism preserving the Tits metric) between two such spaces is induced by a homothety. As an application, we can extend the Mostow and Prasad rigidity theorems to compact singular (orbi)spaces of nonpositive curvature which are homotopy equivalent to a quotient of a symmetric space or Euclidean b…

CAT(0)空间的渐近几何正则性

Asymptotic geometric regularity of CAT(0) spaces

该研究证明了CAT(0)空间在特定条件下与欧几里得空间的双Lipschitz同胚性，并应用此结果得出CAT(1)空间的同胚结论。

Quasiflats in CAT(0) complexes

We show that if X is a piecewise Euclidean 2-complex with a cocompact isometry group, then every 2-quasiflat in X is at finite Hausdorff distance from a subset which is locally flat outside a compact set, and asymptotically conical.

CAT(0) 群作用的有限性

Finiteness of CAT(0) group actions

本文在非正曲率情形下推广黎曼几何经典有限性定理，证明均匀填充 CAT(0)-空间上离散等距群及相应轨道空间的有限性结果，并利用充分塌缩作用的分裂定理给出挠子群阶的通用上界。

Compactification of a diagonal action on the product of CAT(-1) spaces

Let $X$ be a proper, non-compact CAT(-1) space, and $Γ$ a discrete cocompact subgroup of the isometries of $X$. We compactify the diagonal action of $Γ$ on $X \times X$ considering a domain of the horofunction boundary with respect to the maximum metric.

Extending a rigidity result of Lytchak

We extend a rigidity result of Alexander Lytchak by relaxing the requirement that the CAT(1) space be geodesically complete.

Volume pinching theorems for CAT(1) spaces

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 growths under an upper curvature bound.

The unique measure of maximal entropy for a compact rank one locally CAT(0) space

Let $X$ be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. We prove the Bowen-Margulis measure on the space of geodesics is the unique measure of maximal entropy for the geodesic flow, which has topological entropy equal to the critical exponent of the Poincaré series.

Thin actions on CAT(0) spaces

We study groups of isometries of packed, geodesically complete, CAT$(0)$-spaces for which the systole at every point is smaller than a universal constant depending only on the packing, deducing strong rigidity results. We show that if a space as above has some negative curvature behaviour then it cannot support a thin action: this generalizes the classical Margulis Lemma to a broader class of spaces.

可定向CAT(0)空间中可容群的另一种选择及Kazhdan型刚性原理

Torsion subgroups and fixed-point rigidity in CAT(0) geometry

本文证明了有限生成可容群在CAT(0)空间上的作用具有两种可能：要么在具有有界几何的完整CAT(0)空间上存在全局固定点，要么群在R^n上存在无固定点的作用。研究结果对理解可容群在CAT(0)空间上的作用有重要进展。

CAT(0) spaces with boundary the join of two Cantor sets

We will show that if a proper complete CAT(0) space X has a visual boundary homeomorphic to the join of two Cantor sets, and X admits a geometric group action by a group containing a subgroup isomorphic to Z^2, then its Tits boundary is the spherical join of two uncountable discrete sets. If X is geodesically complete, then X is a product, and the group has a finite index subgroup isomorphic to a lattice in the product of two isometry groups of bounded valence bushy trees.

Injective hulls of certain discrete metric spaces and groups

Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. In the 1960es, J. R. Isbell showed that every metric space X has an injective hull E(X). Here it is proved that if X is the vertex set of a connected locally finite graph with a uniform stability property of intervals, then E(X) is a locally finite polyhedral complex with finitely many isometry types of n-cells, isometric to polytopes in l^n_\infty, for each n. This applies to a class of finitely generated groups G, including all word hyperbolic groups and abelian groups, among others. T…

Geometric invariants of spaces with isolated flats

We study those groups that act properly discontinuously, cocompactly, and isometrically on CAT(0) spaces with isolated flats and the Relative Fellow Traveller Property. The groups in question include word hyperbolic CAT(0) groups as well as geometrically finite Kleinian groups and numerous 2-dimensional CAT(0) groups. For such a group we show that there is an intrinsic notion of a quasiconvex subgroup which is equivalent to the inclusion being a quasi-isometric embedding. We also show that the visual boundary of the CAT(0) space is actually an invariant of the group. More generally we show tha…

A characterization of isometries of CAT(0)-space as maps preserving diagonal tube

We give positive answers for questions by Berestovskii. Namely, we prove that every bijection of locally compact geodesically complete and connected at infinity CAT(0)-space $X$ onto itself preserving some fixed distance or satellite relations is an isometry of this space. The proof of this theorem is based on another result stated by Berestovskii as a problem: the metric of the space $X$ may be recovered from its diagonal tube corresponding to an arbitrary number $r > 0$.

Detecting product splittings of CAT(0) spaces

Let $X$ be a proper CAT($0$) space and $G$ a cocompact group of isometries of $X$ without fixed point at infinity. We prove that if $\partial X$ contains an invariant subset of circumradius $π/2$, then $X$ contains a quasi-dense, closed convex subspace that splits as a product. Adding the assumption that the $G$-action on $X$ is properly discontinuous, we give more conditions that are equivalent to a product splitting. In particular, this occurs if $\partial X$ contains a proper nonempty, closed, invariant, $π$-convex set in $\partial X$; or if some nonempty closed, invariant set in $\partial …

The fixed point property and unbounded sets in spaces of negative curvature

Motivated by the well-known cases of the real Hilbert ball and complete R-trees, being both particular cases of CAT(-1) spaces, we give an affirmative answer to the question of whether the geodesically boundedness property is a necessary and sufficient condition for a closed convex subset K of a complete CAT(k) space, with k < 0, to have the fixed point property for nonexpansive mappings

渐近秩处无线性填充的CAT(0)空间

CAT(0) spaces without linear filling at the asymptotic rank

构造了渐近秩为$\nu$的$(\nu+1)$维CAT(0)空间，其积分$\nu$-循环不满足线性等周不等式，填充函数下界为$c v\log v$。

Relations between various boundaries of relatively hyperbolic groups

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 infinity $\partial X$. In this article, we examine the connection between these two spaces at infinity. In particular, we show that $\partial (G,\PP)$ is $G$--equivariantly homeomorphic to the space obtained from $\partial X$ by identifying the peripheral limit points of the sam…

Optimal higher-dimensional Dehn functions for some CAT(0) lattices

Let $X=S\times E \times B$ be the metric product of a symmetric space $S$ of noncompact type, a Euclidean space $E$ and a product $B$ of Euclidean buildings. Let $Γ$ be a discrete group acting isometrically and cocompactly on $X$. We determine a family of quasi-isometry invariants for such $Γ$, namely the $k$-dimensional Dehn functions, which measure the difficulty to fill $k$-spheres by $(k+1)$-balls (for $1\leq k\leq \dim\ X-1$). Since the group $Γ$ is quasi-isometric to the associated CAT(0) space $X$, assertions about Dehn functions for $Γ$ are equivalent tothe corresponding results on fil…

Intrinsic Rank in CAT(0) Spaces

Let $X$ be a proper, geodesically complete CAT(0) space which satisfies Chen and Eberlein's duality condition. We show the existence of a strong notion of rank for $X$ by proving that the parallel sets $P_v$ of geodesics $v$ in $X$ are generically flat. More precisely, let $GX$ be the space of parametrized unit-speed geodesics in $X$. There is a unique $k$ and a dense $G_δ$ set $\mathcal{A}$ in $GX$ such that $P_v$ is isometric to flat Euclidean space $\mathbb{R}^k$, for all $v \in \mathcal{A}$. It follows that $\mathbb{R}^k$ isometrically embeds in $P_v$ for every $v \in GX$.

Packing and doubling in metric spaces with curvature bounded above

We study locally compact, locally geodesically complete, locally CAT(k) spaces (GCBA(k)-spaces). We prove a Croke-type local volume estimate only depending on the dimension of these spaces. We show that a local doubling condition, with respect to the natural measure, implies pure-dimensionality. Then, we consider GCBA(k)-spaces satisfying a uniform packing condition at some fixed scale or a doubling condition at arbitrarily small scale, and prove several compactness results with respect to pointed Gromov-Hausdorff convergence. Finally, as a particular case, we study convergence and stability o…

CAT(-1) 空间上测地流的 Gibbs 测度

Gibbs measures for geodesic flow on CAT(-1) spaces

本文针对 CAT(-1) 空间上的测地流，利用加权拟共形 Patterson 密度构造具有局部乘积结构的 Gibbs 测度，并在有限情形下证明其唯一平衡态性质，且无需势函数在共享线段上取值相同的条件。

正曲率Alexandrov意义下的大$\boldsymbol{\text{CAT}(κ)}$空间中的几何与重心

Geometry and barycenters in large $\CAT(κ)$ spaces

本文研究正曲率Alexandrov意义下大$\text{CAT}(κ)$空间的几何，在重心存在唯一但凸性缺失的情形下，发展相关几何性质并为经验重心等估计量建立有限样本误差界等新统计结果。

Circumcenter extension maps for non-positively curved spaces

We show that every cross ratio preserving homeomorphism between boundaries of Hadamard manifolds extends to a continuous map, called circumcenter extension, provided that the manifolds satisfy certain visibility conditions. We show that this map is a rough isometry, whenever the manifolds admit cocompact group actions by isometries and we improve the quasi-isometry constants provided by Biswas in the case of CAT(-1)} spaces. Finally, we provide a sufficient condition for this map to be an isometry in the case of Hadamard surfaces.

CAT(-1)空间及其他度量Anosov流上测地流的强符号动力学

Strong symbolic dynamics for geodesic flow on CAT(-1) spaces and other metric Anosov flows

该研究证明紧致或凸余紧局部CAT(-1)空间上的测地流可通过带Hölder屋顶函数的不可约有限型移位悬挂流编码，且其技术可推广到射影Anosov表示的测地流，还得到了关于平衡测度的强动力学结果。

次线性Morse边界与准重定向边界的拓扑与动力学性质

Topological and Dynamic Properties of the Sublinearly Morse Boundary and the Quasi-Redirecting Boundary

本文研究配备几何群作用的空间中次线性Morse边界与准重定向边界的拓扑和动力学性质，证明了群作用的极小性、收缩元素的动力学特性及在CAT(0)空间等场景下的相关结论。

On Moebius and conformal maps between boundaries of CAT(-1) spaces

We consider Moebius and conformal homeomorphisms $f : \partial X \to \partial Y$ between boundaries of CAT(-1) spaces $X,Y$ equipped with visual metrics. A conformal map $f$ induces a topological conjugacy of the geodesic flows of $X$ and $Y$, which is flip-equivariant if $f$ is Moebius. We define a function $S(f) : \partial ^2 X \to \mathbb{R}$, the {\it integrated Schwarzian} of $f$, which measures the deviation of the topological conjugacy from being flip-equivariant, in particular vanishing if $f$ is Moebius. Conversely if $X,Y$ are simply connected complete manifolds with pinched negative…

A $K$-quadrilateral cosine characterization of Aleksandrov spaces of curvature bounded above

In this note, we extend the main results of our paper on quasilinearization and curvature of Aleksandrov spaces of curvature $\leq0$ to curvature bounds other than $0$. For non-zero $K$, we employ the previously introduced notion of the $K$-quadrilateral cosine, which is the cosine under parallel transport in model $K$-space, and which is denoted by $\operatorname{cosq}_{K}$. Our principal result states that a geodesically connected metric space (of diameter not greater than $π/\left(2\sqrt{K}\right) $ if $K>0$) is an $\Re_{K}$ domain (otherwise known as a $\operatorname{CAT}\left(K\right) $ s…