Counting subgroups via Mirzakhani's curve counting
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908610189066240 |
|---|---|
| author | Sasaki, Dounnu |
| author_facet | Sasaki, Dounnu |
| contents | Given a hyperbolic surface $Σ$ of genus $g$ with $r$ cusps, Mirzakhani proved that the number of closed geodesics of length at most $L$ and of a given type is asymptotic to $cL^{6g-6+2r}$ for some $c>0$. Since a closed geodesic corresponds to a conjugacy class of the fundamental group $π_1(Σ)$, we extend this to the counting problem of conjugacy classes of finitely generated subgroups of $π_1(Σ)$. Using `half the sum of the lengths of the boundaries of the convex core of a subgroup' instead of the length of a closed geodesic, we prove that the number of such conjugacy classes is similarly asymptotic to $cL^{6g-6+2r}$ for some $c>0$. As a special case, these conjugacy classes can be interpreted as subsurfaces of $Σ$ via their convex cores, and the result can be viewed as counting subsurfaces of a given type. Furthermore, we see that the above length measurement for subgroups is `natural' within the framework of subset currents, which serve as a completion of weighted conjugacy classes of finitely generated subgroups of $π_1(Σ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08109 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting subgroups via Mirzakhani's curve counting Sasaki, Dounnu Geometric Topology 20F34, 30F35 Given a hyperbolic surface $Σ$ of genus $g$ with $r$ cusps, Mirzakhani proved that the number of closed geodesics of length at most $L$ and of a given type is asymptotic to $cL^{6g-6+2r}$ for some $c>0$. Since a closed geodesic corresponds to a conjugacy class of the fundamental group $π_1(Σ)$, we extend this to the counting problem of conjugacy classes of finitely generated subgroups of $π_1(Σ)$. Using `half the sum of the lengths of the boundaries of the convex core of a subgroup' instead of the length of a closed geodesic, we prove that the number of such conjugacy classes is similarly asymptotic to $cL^{6g-6+2r}$ for some $c>0$. As a special case, these conjugacy classes can be interpreted as subsurfaces of $Σ$ via their convex cores, and the result can be viewed as counting subsurfaces of a given type. Furthermore, we see that the above length measurement for subgroups is `natural' within the framework of subset currents, which serve as a completion of weighted conjugacy classes of finitely generated subgroups of $π_1(Σ)$. |
| title | Counting subgroups via Mirzakhani's curve counting |
| topic | Geometric Topology 20F34, 30F35 |
| url | https://arxiv.org/abs/2409.08109 |