# Bi-Lipschitz embeddings of $SRA$-free spaces into Euclidean spaces

## TL;DR

This paper proves that a broad class of metric spaces, including many geometric and algebraic structures, can be embedded into Euclidean spaces using bi-Lipschitz maps, with applications to Alexandrov spaces.

## Contribution

It establishes bi-Lipschitz embedding results for $SRA$-free spaces, including a conjectured embedding for Alexandrov space balls, and introduces an extension theorem for bi-Lipschitz maps.

## Key findings

- $SRA$-free spaces admit bi-Lipschitz embeddings into Euclidean spaces.
- A quantitative embedding theorem for balls in finite-dimensional Alexandrov spaces.
- An extension theorem for bi-Lipschitz maps that may be of independent interest.

## Abstract

$SRA$-free spaces is a wide class of metric spaces including finite dimensional Alexandrov spaces of non-negative curvature, complete Berwald spaces of nonnegative flag curvature, Cayley Graphs of virtually abelian groups and doubling metric spaces of non-positive Busemann curvature with extendable geodesics. This class also includes arbitrary big balls in complete, locally compact $CAT(k)$-spaces $(k \in \mathbb R)$ with locally extendable geodesics, finite-dimensional Alexandrov spaces of curvature $\ge k$ with $k \in R$ and complete Finsler manifolds satisfying the doubling condition. We show that $SRA$-free spaces allow bi-Lipschitz embeddings in Euclidean spaces. As a corollary we obtain a quantitative bi-Lipschitz embedding theorem for balls in finite dimensional Alexandrov spaces of curvature bounded from below conjectured by S. Eriksson-Bique. The main tool of the proof is an extension theorem for bi-Lipschitz maps into Euclidean spaces. This extension theorem is close in nature with the embedding theorem of J. Seo and may be of independent interest.