Quantum isometries and loose embeddings

海森堡群上Wasserstein空间的等距与等距嵌入

Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group

该研究建立欧氏空间与海森堡群最优输运映射的关联，刻画海森堡群上p-Wasserstein空间的测地线结构与度量秩，证明其等距刚性，完善了该类空间的几何性质研究。

(In)equality distance patterns and embeddability into Hilbert spaces

We show that compact Riemannian manifolds, regarded as metric spaces with their global geodesic distance, cannot contain a number of rigid structures such as (a) arbitrarily large regular simplices or (b) arbitrarily long sequences of points equidistant from pairs of points preceding them in the sequence. All of this provides evidence that Riemannian metric spaces admit what we term loose embeddings into finite-dimensional Euclidean spaces: continuous maps that preserve both equality as well as inequality. We also prove a local-to-global principle for Riemannian-metric-space loose embeddabilit…

Equivariant embeddings of trees into hyperbolic spaces

We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of the trees, showing that they are convex-cocompact and asymptotically isometric. On the other hand, focusing on the case of sufficiently transitive groups of automorphisms of locally finite trees, we classify completely all irreducible representations by isometries of hyperbolic spaces. It turns out that in this case our one-parameter family exhausts all non…

有限群作用商空间的粗嵌入

Coarse embeddings of quotients by finite group actions

本文证明了度量空间在有限群等距作用下的商空间保持 Hilbert 空间粗嵌入性质，关键步骤是证明 Hilbert 空间中无序 k 元点组在 1-Wasserstein 距离下可粗嵌入 Hilbert 空间。

渐近-莫比乌斯映射

Asymptotic-Möbius maps

本文提出渐近-莫比乌斯映射，用于研究几何群论中的大型尺度共形性，通过分析点配置的粗略交叉比行为，提供了一种稳定的无穷远处共形性概念，并应用于大型尺度维度和度量余类型的研究。

On quasisymmetric embeddings of the Brownian map and continuum trees

The Brownian map is a model of random geometry on the sphere and as such an important object in probability theory and physics. It has been linked to Liouville Quantum Gravity and much research has been devoted to it. One open question asks for a canonical embedding of the Brownian map into the sphere or other, more abstract, metric spaces. Similarly, Liouville Quantum Gravity has been shown to be "equivalent" to the Brownian map but the exact nature of the correspondence (i.e.\ embedding) is still unknown. In this article we show that any embedding of the Brownian map or continuum random tree…