Bi-Lipschitz Embeddings of Trees into Euclidean Buildings

Abstract

Let $$\mathcal{B}$$ be the affine Bruhat-Tits (or Euclidean) building associated to a semisimple, simply connected linear algebraic group defined over a non-Archimedean local field. We prove that there exist locally finite trees of degree ≥3 which are bi-Lipschitz embedded in $$\mathcal{B}$$ . As a consequence such a Euclidean building cannot be bi-Lipschitz embedded into a Hilbert space.