发表机构
University of Jyväskylä; University of Illinois at Urbana-Champaign（于韦斯屈莱大学; 伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明加倍测地树类无双Lipschitz泛元素，构造反例族并给出超度量空间的正面嵌入结果，否定回答了相关开放问题。
AI 中文摘要
对于$n\ge 3$和$c\in(0,1)$，令$\mathcal{GT}(n,c)$表示价数至多为$n$且分支点以常数$c$均匀相对分离的测地度量树类。我们证明$\mathcal{GT}(n,c)$没有双Lipschitz泛元素。更精确地，我们构造一个族$(T_a)_{a\in[1/4,1/3]}\subset\mathcal{GT}(n,c)$，使得对于每个$n_M\geq 3, c_M\in(0,1)$和每个$M\in\mathcal{GT}(n_M, c_M)$，至多有可数个参数$a$使得$T_a$允许双Lipschitz嵌入到$M$中，而每个$T_a$允许双Lipschitz嵌入到$\mathbb R^2$中。因此障碍既不是维数性的，也不是由平面可嵌入性的失败引起的。这给出了Chrontsios-Garitsis、Ioannidis和Vellis~\cite[问题~1.11]{CGIV2024}的问题的否定答案。此外，我们展示了超度量空间的一个互补的正面结果：每个有界超度量空间$X$允许双Lipschitz嵌入到每个满足$\dim_A X<\dim_{LA} Y$的完备度量空间$Y$中，其中$\dim_A X$和$\dim_{LA} X$分别是Assouad维数和下Assouad维数。
英文摘要
For $n\ge 3$ and $c\in(0,1)$, let $\mathcal{GT}(n,c)$ denote the class of geodesic metric trees of valence at most $n$ whose branch points are uniformly relatively separated with constant $c$. We prove that $\mathcal{GT}(n,c)$ has no bi-Lipschitz universal element. More precisely, we construct a family $(T_a)_{a\in[1/4,1/3]}\subset\mathcal{GT}(n,c)$ such that, for every $n_M\geq 3, c_M\in(0,1)$ and every $M\in\mathcal{GT}(n_M, c_M)$, there are at most countably many parameters $a$ for which $T_a$ admits a bi-Lipschitz embedding into $M$, whereas each $T_a$ admits a bi-Lipschitz embedding into $\mathbb R^2$. Thus the obstruction is neither dimensional nor caused by a failure of planar embeddability. This gives a negative answer to a question of Chrontsios-Garitsis, Ioannidis, and Vellis~\cite[Question~1.11]{CGIV2024}. Furthermore, we show a complementary positive result for ultrametric spaces: every bounded ultrametric space $X$ admits a bi-Lipschitz embedding into every complete metric space $Y$ satisfying $\dim_A X<\dim_{LA} Y$ where $\dim_A X$ and $\dim_{LA} X$ are Assouad and lower Assouad dimensions, respectively.
Comments38 pages, 3 images
