Quantitative bi-Lipschitz embeddings of bounded-curvature manifolds and orbifolds

Abstract

We construct bi-Lipschitz embeddings into Euclidean space for bounded-diameter subsets of manifolds and orbifolds of bounded curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. We also construct global bi-Lipschitz embeddings for spaces of the form ${ℝ}^n∕\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ is a discrete group acting properly discontinuously and by isometries on ${ℝ}^n$. This generalizes results of Naor and Khot. Our approach is based on analyzing the structure of a bounded-curvature manifold at various scales by specializing methods from collapsing theory to a certain class of model spaces. In the process, we develop tools to prove collapsing theory results using algebraic techniques.