#### Volume 22, issue 4 (2018)

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

### Sylvester Eriksson-Bique

Geometry & Topology 22 (2018) 1961–2026
##### 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 ${ℝ}^{n}∕\Gamma$, where $\Gamma$ 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.

##### Keywords
bilipschitz, sectional curvature, Alexandrov, collapsing theory
##### Mathematical Subject Classification 2010
Primary: 30L05, 51F99, 53C21
Secondary: 20H15, 53B20