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.