Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.01801 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909666463711232 |
|---|---|
| author | Matsuda, Ryo Oie, Kanako Shiga, Hiroshige |
| author_facet | Matsuda, Ryo Oie, Kanako Shiga, Hiroshige |
| contents | Let $S$ be an oriented surface of type $(g, n)$. We are interested in geodesics in the curve complex $\mathcal C(S)$ of $S$. In general, two $0$-simplexes in $\mathcal C(S)$ have infinitely many geodesics connecting the two simplexes while another geodesics called tight geodesics are always finitely many. On the other hand, we may find two $0$-simplexes in $\mathcal C(S)$ so that they have only finitely many geodesics between them.
In this paper, we consider the spectrum of the number of geodesics with length $d (\geq 2)$ in $\mathcal C(S)$ and tight geodesics, which is denoted by $\mathfrak{Sp}_d(S)$ and $\mathfrak{Sp}_d^T(S)$, respectively.
In our main theorem, it is shown that $\mathfrak{Sp}_d(S) \subset \mathfrak{Sp}_d^T(S)$ in general, but $\mathfrak{Sp}_2(S)= \mathfrak{Sp}_2^T(S)$. Moreover, we show that $\mathfrak{Sp}_2(S)$ and $\mathfrak{Sp}_2^T(g, n)$ are completely determined in terms of $(g, n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01801 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the spectrum of the number of geodesics and tight geodesics in the curve complex Matsuda, Ryo Oie, Kanako Shiga, Hiroshige Geometric Topology Primary 57K20, Secondary 57K99 Let $S$ be an oriented surface of type $(g, n)$. We are interested in geodesics in the curve complex $\mathcal C(S)$ of $S$. In general, two $0$-simplexes in $\mathcal C(S)$ have infinitely many geodesics connecting the two simplexes while another geodesics called tight geodesics are always finitely many. On the other hand, we may find two $0$-simplexes in $\mathcal C(S)$ so that they have only finitely many geodesics between them. In this paper, we consider the spectrum of the number of geodesics with length $d (\geq 2)$ in $\mathcal C(S)$ and tight geodesics, which is denoted by $\mathfrak{Sp}_d(S)$ and $\mathfrak{Sp}_d^T(S)$, respectively. In our main theorem, it is shown that $\mathfrak{Sp}_d(S) \subset \mathfrak{Sp}_d^T(S)$ in general, but $\mathfrak{Sp}_2(S)= \mathfrak{Sp}_2^T(S)$. Moreover, we show that $\mathfrak{Sp}_2(S)$ and $\mathfrak{Sp}_2^T(g, n)$ are completely determined in terms of $(g, n)$. |
| title | On the spectrum of the number of geodesics and tight geodesics in the curve complex |
| topic | Geometric Topology Primary 57K20, Secondary 57K99 |
| url | https://arxiv.org/abs/2505.01801 |