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