Abstract
We show that the cyclic lamplighter group C 2 ≀ C n embeds into Hilbert space with distortion \(\mathrm{O}(\sqrt{\log n})\) . This matches the lower bound proved by Lee et al. (Geom. Funct. Anal., 2009), answering a question posed in that paper. Thus, the Euclidean distortion of C 2 ≀ C n is \(\varTheta(\sqrt{\log n})\) . Our embedding is constructed explicitly in terms of the irreducible representations of the group. Since the optimal Euclidean embedding of a finite group can always be chosen to be equivariant, as shown by Aharoni et al. (Isr. J. Math. 52(3):251–265, 1985) and by Gromov (see de Cornulier et. al. in Geom. Funct. Anal., 2009), such representation-theoretic considerations suggest a general tool for obtaining upper and lower bounds on Euclidean embeddings of finite groups.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aharoni, I., Maurey, B., Mityagin, B.S.: Uniform embeddings of metric spaces and of Banach spaces into Hilbert spaces. Isr. J. Math. 52(3), 251–265 (1985)
Alon, N., Lubotzky, A., Wigderson, A.: Semi-direct product in groups and zig-zag product in graphs: connections and applications (extended abstract). In: 42nd IEEE Symposium on Foundations of Computer Science, Las Vegas, NV, 2001, pp. 630–637. IEEE Computer Society, Los Alamitos (2001)
Alon, N., Roichman, Y.: Random Cayley graphs and expanders. Random Struct. Algorithms 5(2), 271–284 (1994)
Arora, S., Lee, J.R., Naor, A.: Euclidean distortion and the sparsest cut. J. Am. Math. Soc. 21(1), 1–21 (2008)
Ball, K.: Markov chains, Riesz transforms and Lipschitz maps. Geom. Funct. Anal. 2(2), 137–172 (1992)
Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, vol. 1. American Mathematical Society Colloquium Publications, vol. 48. American Mathematical Society, Providence (2000)
Bourgain, J.: The metrical interpretation of superreflexivity in Banach spaces. Isr. J. Math. 56(2), 222–230 (1986)
Cheeger, J., Kleiner, B.: Differentiating maps into L 1 and the geometry of BV functions. Preprint (2006)
de Cornulier, Y., Tessera, R., Valette, A.: Isometric group actions on Hilbert spaces: growth of cocycles. Geom. Funct. Anal. 17, 770–792 (2007)
Fulton, W., Harris, J.: Representation Theory. Graduate Texts in Mathematics, vol. 129. Springer, New York (1991). A first course, Readings in Mathematics
Khot, S., Naor, A.: Nonembeddability theorems via Fourier analysis. Math. Ann. 334(4), 821–852 (2006)
Khot, S., Vishnoi, N.K.: Integrability gap for cut problems and embeddability of negative type metrics into l1. In: 46th Annual Symposium on Foundations of Computer Science, pp. 53–62. ACM, New York (2005)
Lee, J.R., Naor, A.: L p metrics on the Heisenberg group and the Goemans–Linial conjecture. In: 47th Annual Symposium on Foundations of Computer Science, pp. 99–108. ACM, New York (2006)
Lee, J.R., Naor, A., Peres, Y.: Trees and Markov convexity. Geom. Funct. Anal. 18(5), 1609–1659 (2008)
Lyons, R., Pemantle, R., Peres, Y.: Random walks on the lamplighter group. Ann. Probab. 24(4), 1993–2006 (1996)
Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)
Naor, A., Peres, Y., Schramm, O., Sheffield, S.: Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J. 134(1), 165–197 (2006)
Reingold, O., Vadhan, S., Wigderson, A.: Entropy waves, the zig-zag graph product, and new constant-degree expanders. Ann. Math. (2) 155(1), 157–187 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was conducted while T. Austin was visiting the Courant Institute of Mathematical Sciences, New York University.
A. Naor’s research supported by NSF grants CCF-0635078 and DMS-0528387.
Rights and permissions
About this article
Cite this article
Austin, T., Naor, A. & Valette, A. The Euclidean Distortion of the Lamplighter Group. Discrete Comput Geom 44, 55–74 (2010). https://doi.org/10.1007/s00454-009-9162-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-009-9162-6