Geometry & Topology

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

Sylvester Eriksson-Bique

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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 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1523584816

Digital Object Identifier
doi:10.2140/gt.2018.22.1961

Mathematical Reviews number (MathSciNet)
MR3784515

Zentralblatt MATH identifier
06864331

Subjects
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

Keywords
bilipschitz sectional curvature Alexandrov collapsing theory

Citation

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. https://projecteuclid.org/euclid.gt/1523584816


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