On Salem numbers of degree 4 and arithmetic hyperbolic orbifolds
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912725906489344 |
|---|---|
| author | Dória, Cayo Murillo, Plinio G. P. |
| author_facet | Dória, Cayo Murillo, Plinio G. P. |
| contents | In this article, we construct an arithmetic hyperbolic $6-$orbifold $\mathcal{O}$ such that, any square-rootable Salem number of degree at most $4$ over $\mathbb{Q}$ is realized as the exponential of the length of a closed geodesic in $\mathcal{O}$. We also prove that $n=6$ is the minimal dimension among arithmetic hyperbolic orbifolds of the first type where it can be obtained. In an appendix, we establish a general relation between the discriminant of a Salem number and the determinant of a quadratic space which realizes it. In particular, for any $m,d>0$ we present a geometric proof of the existence of Salem numbers of degree $2m$ with discriminant $(-1)^{m+1}d$ in $\mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17041 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Salem numbers of degree 4 and arithmetic hyperbolic orbifolds Dória, Cayo Murillo, Plinio G. P. Number Theory Group Theory Geometric Topology In this article, we construct an arithmetic hyperbolic $6-$orbifold $\mathcal{O}$ such that, any square-rootable Salem number of degree at most $4$ over $\mathbb{Q}$ is realized as the exponential of the length of a closed geodesic in $\mathcal{O}$. We also prove that $n=6$ is the minimal dimension among arithmetic hyperbolic orbifolds of the first type where it can be obtained. In an appendix, we establish a general relation between the discriminant of a Salem number and the determinant of a quadratic space which realizes it. In particular, for any $m,d>0$ we present a geometric proof of the existence of Salem numbers of degree $2m$ with discriminant $(-1)^{m+1}d$ in $\mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}$. |
| title | On Salem numbers of degree 4 and arithmetic hyperbolic orbifolds |
| topic | Number Theory Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2510.17041 |