Geometry & Topology

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

Sylvester Eriksson-Bique

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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 n Γ , where Γ is a discrete group acting properly discontinuously and by isometries on 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.

Article information

Geom. Topol., Volume 22, Number 4 (2018), 1961-2026.

Received: 21 October 2015
Revised: 3 July 2017
Accepted: 31 July 2017
First available in Project Euclid: 13 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30L05: Geometric embeddings of metric spaces 51F99: None of the above, but in this section 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
Secondary: 20H15: Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25] 53B20: Local Riemannian geometry

bilipschitz sectional curvature Alexandrov collapsing theory


Eriksson-Bique, Sylvester. Quantitative bi-Lipschitz embeddings of bounded-curvature manifolds and orbifolds. Geom. Topol. 22 (2018), no. 4, 1961--2026. doi:10.2140/gt.2018.22.1961.

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