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

Eriksson-Bique, Sylvester

doi:10.2140/gt.2018.22.1961

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

Sylvester Eriksson-Bique

Geometry & Topology 22 (2018) 1961–2026

DOI: 10.2140/gt.2018.22.1961

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 $ℝ^{n} ∕ Γ$ , where $Γ$ 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

References

Publication

Received: 21 October 2015

Revised: 3 July 2017

Accepted: 31 July 2017

Published: 5 April 2018

Proposed: Yasha Eliashberg

Seconded: Dmitri Burago, Bruce Kleiner

Authors

Sylvester Eriksson-Bique
Department of Mathematics UCLA Los Angeles, CA United States

Volume 22, issue 4 (2018)