## Abstract

Let G be a finite group of order n and for 1 ≤ i≤ k+ 1 let V_{i}= { i} × G . Viewing each V_{i} as a 0-dimensional complex, let Y_{G}_{,}_{k} denote the simplicial join V_{1}∗ ⋯ ∗ V_{k}_{+}_{1} . For A⊂ G let Y_{A}_{,}_{k} be the subcomplex of Y_{G}_{,}_{k} that contains the (k- 1) -skeleton of Y_{G}_{,}_{k} and whose k-simplices are all { (1 , x_{1}) , … , (k+ 1 , x_{k}_{+}_{1}) } ∈ Y_{G}_{,}_{k} such that x_{1}⋯ x_{k}_{+}_{1}∈ A . Let L_{k}_{-}_{1} denote the reduced (k- 1) -th Laplacian of Y_{A}_{,}_{k} , acting on the space C^{k}^{-}^{1}(Y_{A}_{,}_{k}) of real valued (k- 1) -cochains of Y_{A}_{,}_{k} . The (k- 1) -th spectral gap μ_{k}_{-}_{1}(Y_{A}_{,}_{k}) of Y_{A}_{,}_{k} is the minimal eigenvalue of L_{k}_{-}_{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 language | English |
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Journal | Journal of Applied and Computational Topology |

DOIs | |

State | Accepted/In press - 2023 |

## Keywords

- High dimensional Laplacians
- Random complexes
- Spectral gap

## All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Applied Mathematics
- Geometry and Topology