A central limit theorem for connected components of random coverings of manifolds with nilpotent fundamental groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914421439201280 |
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| author | Abdesselam, Abdelmalek |
| author_facet | Abdesselam, Abdelmalek |
| contents | There is a well understood way of generating random coverings of a fixed manifold by sampling homomorphisms from the fundamental group of this manifold into the symmetric group. We prove a central limit theorem for the number of connected components of these random coverings when the fundamental group is nilpotent. This provides a nonabelian generalization of an earlier result by the author and Shannon Starr in the case of the torus where the fundamental group is a free abelian group of rank at least two. Our result relies on the work of du Sautoy and Grunewald on the subgroup growth zeta functions of nilpotent groups, and on Delange's generalization of the Wiener-Ikehara Tauberian theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24499 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A central limit theorem for connected components of random coverings of manifolds with nilpotent fundamental groups Abdesselam, Abdelmalek Probability High Energy Physics - Theory Group Theory Geometric Topology Number Theory 05A16, 11M41, 20E07, 20F65, 20P05, 57M10, 60F05 There is a well understood way of generating random coverings of a fixed manifold by sampling homomorphisms from the fundamental group of this manifold into the symmetric group. We prove a central limit theorem for the number of connected components of these random coverings when the fundamental group is nilpotent. This provides a nonabelian generalization of an earlier result by the author and Shannon Starr in the case of the torus where the fundamental group is a free abelian group of rank at least two. Our result relies on the work of du Sautoy and Grunewald on the subgroup growth zeta functions of nilpotent groups, and on Delange's generalization of the Wiener-Ikehara Tauberian theorem. |
| title | A central limit theorem for connected components of random coverings of manifolds with nilpotent fundamental groups |
| topic | Probability High Energy Physics - Theory Group Theory Geometric Topology Number Theory 05A16, 11M41, 20E07, 20F65, 20P05, 57M10, 60F05 |
| url | https://arxiv.org/abs/2603.24499 |