Expansion of Integer Matrices over Various Rings
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909799663271936 |
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| author | Szymański, Jakub |
| author_facet | Szymański, Jakub |
| contents | In this article, we explore the problem of constructing high-dimensional expanders through the study of relations between expansion constants over different rings. We investigate expansion constants of integer matrices regarded as morphisms between free modules over $\mathbb{R}$, $\mathbb{Z}$, and $\mathbb{Z}/p\mathbb{Z}$. We introduce a new condition which we call integral spanning regarding kernels of integer matrices, and prove that it ensures equality of real and integral expansions. In addition, we prove a bound on expansion constants over finite fields for a certain class of matrices in terms of the corresponding integral expansions. As an application, one may use this theorem to bound the expansion of codifferentials over $\mathbb{Z}/2\mathbb{Z}$ in degrees $0$ and $1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17823 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Expansion of Integer Matrices over Various Rings Szymański, Jakub Group Theory 20F65 In this article, we explore the problem of constructing high-dimensional expanders through the study of relations between expansion constants over different rings. We investigate expansion constants of integer matrices regarded as morphisms between free modules over $\mathbb{R}$, $\mathbb{Z}$, and $\mathbb{Z}/p\mathbb{Z}$. We introduce a new condition which we call integral spanning regarding kernels of integer matrices, and prove that it ensures equality of real and integral expansions. In addition, we prove a bound on expansion constants over finite fields for a certain class of matrices in terms of the corresponding integral expansions. As an application, one may use this theorem to bound the expansion of codifferentials over $\mathbb{Z}/2\mathbb{Z}$ in degrees $0$ and $1$. |
| title | Expansion of Integer Matrices over Various Rings |
| topic | Group Theory 20F65 |
| url | https://arxiv.org/abs/2509.17823 |