## Geometry & Topology

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

Sylvester Eriksson-Bique

#### 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.

#### Article information

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

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

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

#### 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

#### References

• S Alexander, V Kapovitch, A Petrunin, Alexandrov geometry, in preparation
• A Andoni, A Naor, O Neiman, Snowflake universality of Wasserstein spaces, preprint (2015)
• P Assouad, Plongements lipschitziens dans ${\bf R}\sp{n}$, Bull. Soc. Math. France 111 (1983) 429–448
• L Auslander, Bieberbach's theorems on space groups and discrete uniform subgroups of Lie groups, Ann. of Math. 71 (1960) 579–590
• M Bonk, U Lang, Bi-Lipschitz parameterization of surfaces, Math. Ann. 327 (2003) 135–169
• J Bourgain, On Lipschitz embedding of finite metric spaces in Hilbert space, Israel J. Math. 52 (1985) 46–52
• D Burago, Y Burago, S Ivanov, A course in metric geometry, Graduate Studies in Mathematics 33, Amer. Math. Soc., Providence, RI (2001)
• Y Burago, M Gromov, G Perel'man, A D Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992) 3–51 In Russian; translated in Russian Math. Surveys 47 (1992) 1–58
• P Buser, H Karcher, The Bieberbach case in Gromov's almost flat manifold theorem, from “Global differential geometry and global analysis” (D Ferus, W Kühnel, U Simon, B Wegner, editors), Lecture Notes in Math. 838, Springer (1981) 82–93
• P Buser, H Karcher, Gromov's almost flat manifolds, Astérisque 81, Soc. Math. France, Paris (1981)
• J Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999) 428–517
• J Cheeger, K Fukaya, M Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992) 327–372
• M J Collins, On Jordan's theorem for complex linear groups, J. Group Theory 10 (2007) 411–423
• L J Corwin, F P Greenleaf, Representations of nilpotent Lie groups and their applications, I: Basic theory and examples, Cambridge Studies in Advanced Mathematics 18, Cambridge Univ. Press (1990)
• Y Ding, A restriction for singularities on collapsing orbifolds, ISRN Geom. (2011) art. id. 502814, 7 pages
• Y Ding, A restriction for singularities on collapsing orbifolds, preprint (2011)
• K Fukaya, A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom. 28 (1988) 1–21
• K Fukaya, Collapsing Riemannian manifolds to ones with lower dimension, II, J. Math. Soc. Japan 41 (1989) 333–356
• K Fukaya, Hausdorff convergence of Riemannian manifolds and its applications, from “Recent topics in differential and analytic geometry” (T Ochiai, editor), Adv. Stud. Pure Math. 18, Academic Press, Boston (1990) 143–238
• K Fukaya, Collapsing Riemannian manifolds and its applications, from “Proceedings of the International Congress of Mathematicians, I” (I Satake, editor), Math. Soc. Japan, Tokyo (1991) 491–500
• P Ghanaat, M Min-Oo, E A Ruh, Local structure of Riemannian manifolds, Indiana Univ. Math. J. 39 (1990) 1305–1312
• R E Greene, H Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 131 (1988) 119–141
• M Gromov, Almost flat manifolds, J. Differential Geom. 13 (1978) 231–241
• M Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics 152, Birkhäuser, Boston (1999)
• K Grove, H Karcher, How to conjugate $C\sp{1}$\!–close group actions, Math. Z. 132 (1973) 11–20
• I Haviv, O Regev, The Euclidean distortion of flat tori, J. Topol. Anal. 5 (2013) 205–223
• J Heinonen, Lectures on analysis on metric spaces, Springer (2001)
• J Heinonen, Lectures on Lipschitz analysis, Report, Matematiikan Laitos 100, Univ. of Jyväskylä (2005)
• J Heinonen, S Keith, Flat forms, bi-Lipschitz parameterizations, and smoothability of manifolds, Publ. Math. Inst. Hautes Études Sci. 113 (2011) 1–37
• P Indyk, J Matoušek, from “Handbook of discrete and computational geometry” (J E Goodman, J O'Rourke, editors), 2nd edition, Chapman & Hall/CRC, Boca Raton, FL (2004) 177–196
• V Kapovitch, Perelman's stability theorem, from “Surveys in differential geometry, XI” (J Cheeger, K Grove, editors), International Press, Somerville, MA (2007) 103–136
• H Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30 (1977) 509–541
• S Khot, A Naor, Nonembeddability theorems via Fourier analysis, Math. Ann. 334 (2006) 821–852
• B Kleiner, J Lott, Locally collapsed $3$–manifolds, Astérisque 365, Soc. Math. France, Paris (2014) 7–99
• T J Laakso, Ahlfors $Q$–regular spaces with arbitrary $Q>1$ admitting weak Poincaré inequality, Geom. Funct. Anal. 10 (2000) 111–123
• U Lang, C Plaut, Bilipschitz embeddings of metric spaces into space forms, Geom. Dedicata 87 (2001) 285–307
• U Lang, V Schroeder, Kirszbraun's theorem and metric spaces of bounded curvature, Geom. Funct. Anal. 7 (1997) 535–560
• K Luosto, Ultrametric spaces bi-Lipschitz embeddable in ${\bf R}^n$, Fund. Math. 150 (1996) 25–42
• J Luukkainen, H Movahedi-Lankarani, Minimal bi-Lipschitz embedding dimension of ultrametric spaces, Fund. Math. 144 (1994) 181–193
• A I Mal'cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat. 13 (1949) 9–32 In Russian; translated in Amer. Math. Soc. Trans. 39 (1951), 33 pages
• J Matoušek, Bi-Lipschitz embeddings into low-dimensional Euclidean spaces, Comment. Math. Univ. Carolin. 31 (1990) 589–600
• J Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976) 293–329
• A Nagel, E M Stein, S Wainger, Balls and metrics defined by vector fields, I: Basic properties, Acta Math. 155 (1985) 103–147
• A Naor, An introduction to the Ribe program, Jpn. J. Math. 7 (2012) 167–233
• A Naor, O Neiman, Assouad's theorem with dimension independent of the snowflaking, Rev. Mat. Iberoam. 28 (2012) 1123–1142
• A Naor, Y Peres, O Schramm, S Sheffield, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Math. J. 134 (2006) 165–197
• I G Nikolaev, Bounded curvature closure of the set of compact Riemannian manifolds, Bull. Amer. Math. Soc. 24 (1991) 171–177
• P Pansu, Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. 129 (1989) 1–60
• J G Ratcliffe, Foundations of hyperbolic manifolds, 2nd edition, Graduate Texts in Mathematics 149, Springer (2006)
• M Romney, Conformal Grushin spaces, Conform. Geom. Dyn. 20 (2016) 97–115
• X Rong, On the fundamental groups of manifolds of positive sectional curvature, Ann. of Math. 143 (1996) 397–411
• E A Ruh, Almost flat manifolds, J. Differential Geom. 17 (1982) 1–14
• S Semmes, Bi-Lipschitz mappings and strong $A_\infty$ weights, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993) 211–248
• S Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about $A_\infty$–weights, Rev. Mat. Iberoamericana 12 (1996) 337–410
• J Seo, A characterization of bi-Lipschitz embeddable metric spaces in terms of local bi-Lipschitz embeddability, Math. Res. Lett. 18 (2011) 1179–1202
• C Sormani, How Riemannian manifolds converge, from “Metric and differential geometry” (X Dai, X Rong, editors), Progr. Math. 297, Springer (2012) 91–117
• T Tao, Hilbert's fifth problem and related topics, Graduate Studies in Mathematics 153, Amer. Math. Soc., Providence, RI (2014)
• W P Thurston, The geometry and topology of three-manifolds, lecture notes, Princeton University (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m {\unhbox0
• W P Thurston, Three-dimensional geometry and topology, I, Princeton Mathematical Series 35, Princeton Univ. Press (1997)
• T Toro, Surfaces with generalized second fundamental form in $L^2$ are Lipschitz manifolds, J. Differential Geom. 39 (1994) 65–101
• T Toro, Geometric conditions and existence of bi-Lipschitz parameterizations, Duke Math. J. 77 (1995) 193–227
• N T Varopoulos, L Saloff-Coste, T Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics 100, Cambridge Univ. Press (1992)