L₁ and L₂ embeddings of the symmetric group

2025/12/10 by Cosmas Kravaris, Kravaris, Cosmas
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Geometric and Algebraic Topology #Group Theory (math.GR) #Limits and Structures in Graph Theory #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2512.09226

openalex publication_date 2025/12/10 · openalex created_date 2025/12/12 · openalex updated_date 2026/07/28

Abstract

We show that the Cayley graph of the symmetric group Sym_n generated by the cycle (123...n) and the transposition (12) embeds into L₁ with bi-Lipschitz distortion O(1). This answers a question of Ostrovskii, and along with Kassabov's theorem gives the first example of a sequence of groups which embed bi-Lipschitzly into L₁ for one choice of bounded size generating sets, but not for another choice of bounded size generating sets. In particular, the Cayley graphs generated by the cycle and the transposition cannot contain coarsely any unbounded sequence of expander graphs. Moreover, within the context of the Ribe program, they are a new example of bounded degree Cayley graphs which are test spaces for Rademacher type.

Citations

• Nearly Optimal Embeddings of Flat Tori
• Median geometry for spaces with measured walls and for groups
• An introduction to the Ribe program
• Embeddings of discrete groups and the speed of random walks
• The Euclidean distortion of the lamplighter group
• Nonembeddability theorems via Fourier analysis
• Rate of Escape of the Mixer Chain
• Symmetric Groups and Expander Graphs
• Metrics on diagram groups and uniform embeddings in a Hilbert space

Related

• Symmetric Groups and Expanders
• Symmetric Groups and Expander Graphs
• Cayley graph on symmetric groups with generating block transposition sets
• A note on a Cayley graph of S_n
• Automorphism groups of Cayley graphs generated by general transposition sets