Devin Murray, Yulan Qing, Abdul Zalloum
We introduce the notion of k-lower divergence for geodesic rays in CAT(0) spaces. Building on the work of Charney and Sultan we give various characterizations of k-contracting geodesic rays using k-lower divergence and k-slim triangles. We also characterize k-contracting geodesic rays in CAT(0) cube complexes using sequences of well-separated hyperplanes.
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We show that if G is an admissible group acting geometrically on a CAT(0) space X, then G is a hierarchically hyperbolic space and with mild assumptions the sublinearly-Morse boundary of the group is a topological model for associated Poisson boundaries .
Genericity of sublinearly Morse directions in general metric spaces
本文研究了在一般度量空间中亚线性Morse方向的普遍性，证明其在horofunction边界中具有完整的Patterson-Sullivan测度。
Given a sublinear function $κ$, $κ$-Morse boundaries $\pka X$ of proper \CAT spaces are introduced by Qing, Rafi and Tiozzo. It is a topological space that consists of a large set of quasi-geodesic rays and it is quasi-isometrically invariant and metrizable. In this paper, we study the sublinearly Morse boundaries with the assumption that there is a proper cocompact action of a group $G$ on the \CAT space in question. We show that $G$ acts minimally on $\pka G$ and that contracting elements of $G$ induces a weak north-south dynamic on $\pka G$. Furthermore, we show that a homeomorphism $f \fro…
We prove that the sublinearly Morse boundary of every known cubulated group continuously injects in the Gromov boundary of a certain hyperbolic graph. We also show that for all CAT(0) cube complexes, convergence to sublinearly Morse geodesic rays has a simple combinatorial description using the hyperplanes crossed by such sequences. As an application of this combinatorial description, we show that a certain subspace of the Roller boundary continously surjects on the subspace of the visual boundary consisting of sublinearly Morse geodesic rays.
We show that the sublinearly Morse boundary of a CAT(0) cubical group with a factor system is well-defined up to homeomorphism with respect to the visual topology. The key tool used in the proof is a new topology on sublinearly Morse boundaries that is induced by group actions on hyperbolic spaces that are sufficiently nice, for example, largest acylindrical actions. Using the same techniques, we obtain a explicit description of this new topology on the sublinearly Morse boundary of any hierarchically hyperbolic group in terms of medians. Finally, we explicitly describe the sublinear Morse bou…
Sublinearly Morseness in Higher Rank Symmetric Spaces
本文在非紧型高阶对称空间中发展了“次线性Morse边界”理论，并证明了相应的次线性Morse引理。
We show that the sublinearly Morse directions in the visual boundary of a rank-1 CAT(0) space with a geometric group action are generic in several commonly studied senses of the word, namely with respect to Patterson-Sullivan measures and stationary measures for random walks. We deduce that the sublinearly Morse boundary is a model of the Poisson boundary for finitely supported random walks on groups acting geometrically on rank-1 CAT (0) spaces. We prove an analogous result for mapping class group actions on Teichmüller space. Our main technical tool is a criterion, valid in any unique geodes…
A sublinear biLipschitz equivalence (SBE) between metric spaces is a map from one space to another that distorts distances with bounded multiplicative constants and sublinear additive error. Given any sublinear function $κ$, $κ$-Morse boundaries are defined for all geodesic proper metric spaces as a quasi-isometrically invariant and metrizable topological space of quasi-geodesic rays. In this paper, we prove that $κ$-Morse boundaries of proper geodesic metric spaces are invariant under suitable SBEs. A tool in the proof is the use of sublinear rays, that is, sublinear bilispchitz embeddings of…
We introduce and begin a systematic study of sublinearly contracting projections. We give two characterizations of Morse quasi-geodesics in an arbitrary geodesic metric space. One is that they are sublinearly contracting; the other is that they have completely superlinear divergence. We give a further characterization of sublinearly contracting projections in terms of projections of geodesic segments.
Curtain Characterization of Sublinearly Morse Geodesics in CAT(0) Spaces
本文证明每个 CAT(0) 空间的次线性 Morse 边界可连续嵌入某双曲空间的 Gromov 边界，利用帘幕机制给出多种刻画，并回答了 Petyt-Spriano-Zalloum 的多个问题。
To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. Gromov showed that quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion of the boundary of a hyperbolic group. Croke and Kleiner showed that the visual boundary of non-positively curved (CAT(0)) groups is not well-defined, since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. For any sublinear function $κ$, we consider a subset of the visual boundary called the $κ$-Morse boundary and show that …
Sublinearly Morse Boundary II: Proper geodesic spaces
本文为正常测地度量空间（包括有限生成群）构造了亚线性 Morse 边界，该边界拟等距不变且可度量，并应用于映射类群和相对双曲群的 Poisson 边界实现。
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。
Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinearly Morse boundary and proved that this boundary is a quasi-isometry invariant which captures a notion of generic direction in a broad context. In this article, we develop the geometric foundations of sublinear Morseness in the mapping class group and Teichmüller space. We prove that their sublinearly Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the…
The quasi-redirecting Boundary
本研究将Gromov边界推广为拟重定向边界，适用于更广的度量空间，具有拟等距不变性，与多种已知边界相容，并完整描述了Croke-Kleiner群的该边界，发现了新的QI不变类Morse拟测地线。
Atypical generic directions in Teichmüller space
本文受映射类群随机游走通用方向的几何刻画问题启发，利用 Chaika–Masur–Wolf 与 Durham–Zalloum 的工作，首次构造出具有极小非唯一遍历垂直叶状结构的次线性 Morse Teichmüller 测地射线。
Ancona inequalities along generic geodesic rays
本文针对非顺从群上的随机游走提出了多个版本的 Ancona 不等式，通过在具有狭窄点的 Morse 子集和按比例收缩射线上建立不等式，实现了边界嵌入，并推广至具有 Morse 超平面的 CAT(0) 立方复合体。
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