Expanding groups with large diameter
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917274628128768 |
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| author | Eberhard, Sean Sabatini, Luca |
| author_facet | Eberhard, Sean Sabatini, Luca |
| contents | We study how the spectral gap and diameter of Cayley graphs depend strongly on the choice of generating set.
We answer a question of Pyber and Szabó (2013)
by exhibiting a sequence of finite groups $G_n$ with $|G_n| \to \infty$ admitting bounded generating sets $X_n,Y_n$ such that $\operatorname{Cay}(G_n,X_n)$ is an expander
while $\operatorname{Cay}(G_n,Y_n)$ has super-polylogarithmic diameter.
The construction uses the semidirect product $G_n = C_p^{n-1} \rtimes S_n$ with $p$ exponentially large in $n$,
and the analysis reduces to bounding some exponential sums of permutational type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13582 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Expanding groups with large diameter Eberhard, Sean Sabatini, Luca Group Theory Combinatorics Number Theory Primary 20F65, Secondary 05C25, 11L07 We study how the spectral gap and diameter of Cayley graphs depend strongly on the choice of generating set. We answer a question of Pyber and Szabó (2013) by exhibiting a sequence of finite groups $G_n$ with $|G_n| \to \infty$ admitting bounded generating sets $X_n,Y_n$ such that $\operatorname{Cay}(G_n,X_n)$ is an expander while $\operatorname{Cay}(G_n,Y_n)$ has super-polylogarithmic diameter. The construction uses the semidirect product $G_n = C_p^{n-1} \rtimes S_n$ with $p$ exponentially large in $n$, and the analysis reduces to bounding some exponential sums of permutational type. |
| title | Expanding groups with large diameter |
| topic | Group Theory Combinatorics Number Theory Primary 20F65, Secondary 05C25, 11L07 |
| url | https://arxiv.org/abs/2602.13582 |