On universal elements for doubling geodesic trees

Abstract

For $n\ge 3$ and $c\in (0,1)$, let $\mathcal{G}\mathcal{T}(n,c)$ 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 $\mathcal{G}\mathcal{T}(n,c)$ has no bi-Lipschitz universal element. More precisely, we construct a family $(T_a{)}_{a\in [1/4,1/3]}\subset \mathcal{G}\mathcal{T}(n,c)$ such that, for every $n_M\ge 3,c_M\in (0,1)$ and every $M\in \mathcal{G}\mathcal{T}(n_M,c_M)$, 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.

38 pages, 3 images