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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2601.02710 |
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| _version_ | 1866908749618216960 |
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| author | Luo, Qiliang |
| author_facet | Luo, Qiliang |
| contents | Let $M$ and $N$ be two closed hyperbolic Riemann surfaces. The Ehrenpreis Conjecture (proved by Kahn-Markovic) asserts that for any $ε>0$ there are finite covers $M_ε\to M$, and $N_ε\to N$, such that the Teichmuller distance (in the suitable moduli space) between $M_ε$ and $N_ε$ is less than $ε$. It is natural to ask how large the degrees of these coverings need to be to achieve that the distance between $M_ε$ and $N_ε$ is less than $ε$. In this paper we show that there exists a constant $k>0$, depending only on $M$ and $N$, so that the covers $M_ε\to M$, and $N_ε\to N$, can be chosen to have the degrees less than $ε^{-k}$. We show that this bound is optimal by considering the case when $M$ and $N$ are arithmetic Riemann surfaces with the same invariant trace field which are not commensurable to each other. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02710 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Effective Ehrenpreis Conjecture Luo, Qiliang Geometric Topology Let $M$ and $N$ be two closed hyperbolic Riemann surfaces. The Ehrenpreis Conjecture (proved by Kahn-Markovic) asserts that for any $ε>0$ there are finite covers $M_ε\to M$, and $N_ε\to N$, such that the Teichmuller distance (in the suitable moduli space) between $M_ε$ and $N_ε$ is less than $ε$. It is natural to ask how large the degrees of these coverings need to be to achieve that the distance between $M_ε$ and $N_ε$ is less than $ε$. In this paper we show that there exists a constant $k>0$, depending only on $M$ and $N$, so that the covers $M_ε\to M$, and $N_ε\to N$, can be chosen to have the degrees less than $ε^{-k}$. We show that this bound is optimal by considering the case when $M$ and $N$ are arithmetic Riemann surfaces with the same invariant trace field which are not commensurable to each other. |
| title | The Effective Ehrenpreis Conjecture |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2601.02710 |