Random balanced Cayley complexes

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Abstract

Let G be a finite group of order n and for 1 ≤ i≤ k+ 1 let Vi= { i} × G . Viewing each Vi as a 0-dimensional complex, let YG,k denote the simplicial join V1∗ ⋯ ∗ Vk+1 . For A⊂ G let YA,k be the subcomplex of YG,k that contains the (k- 1) -skeleton of YG,k and whose k-simplices are all { (1 , x1) , … , (k+ 1 , xk+1) } ∈ YG,k such that x1⋯ xk+1∈ A . Let Lk-1 denote the reduced (k- 1) -th Laplacian of YA,k , acting on the space Ck-1(YA,k) of real valued (k- 1) -cochains of YA,k . The (k- 1) -th spectral gap μk-1(YA,k) of YA,k is the minimal eigenvalue of Lk-1 . The following k-dimensional analogue of the Alon–Roichman theorem is proved: Let k≥ 1 and ϵ> 0 be fixed and let A be a random subset of G of size m=⌈9k2logDϵ2⌉ where D is the sum of the degrees of the complex irreducible representations of G. Then Pr[μk-1(YA,k)<(1-ϵ)m]=O(1n).

Original languageEnglish
JournalJournal of Applied and Computational Topology
DOIs
StateAccepted/In press - 2023

Keywords

  • High dimensional Laplacians
  • Random complexes
  • Spectral gap

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics
  • Geometry and Topology

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