The non-peripheral curve graph and divergence in big mapping class groups
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917357719388160 |
|---|---|
| author | Bar-Natan, Assaf Qing, Yulan Rafi, Kasra |
| author_facet | Bar-Natan, Assaf Qing, Yulan Rafi, Kasra |
| contents | We introduce a numerical invariant $ζ(Σ)$ measuring the end-complexity of $Σ$ and use it to organize coarse-geometric features of Map($Σ$). Our main tool is the \emph{non-peripheral curve graph} $C_{\rm np}(Σ)$, whose vertices are those essential simple closed curves that cannot be pushed out of every compact subsurface, with edges given by disjointness. Assuming Map($Σ$) is CB-generated and $ζ(Σ)\ge 5$, we prove that $C_{\rm np}(Σ)$ is connected, has infinite diameter, is Gromov hyperbolic, and that the Map($Σ$)-action has unbounded orbits. As applications, we show that if $ζ(Σ)\ge 4$ then Map($Σ$) has infinite coarse rank, and if $ζ(Σ)\ge 5$ then Map($Σ$) has at most quadratic divergence, hence is one-ended. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21560 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The non-peripheral curve graph and divergence in big mapping class groups Bar-Natan, Assaf Qing, Yulan Rafi, Kasra Geometric Topology 20F65, 57K20 We introduce a numerical invariant $ζ(Σ)$ measuring the end-complexity of $Σ$ and use it to organize coarse-geometric features of Map($Σ$). Our main tool is the \emph{non-peripheral curve graph} $C_{\rm np}(Σ)$, whose vertices are those essential simple closed curves that cannot be pushed out of every compact subsurface, with edges given by disjointness. Assuming Map($Σ$) is CB-generated and $ζ(Σ)\ge 5$, we prove that $C_{\rm np}(Σ)$ is connected, has infinite diameter, is Gromov hyperbolic, and that the Map($Σ$)-action has unbounded orbits. As applications, we show that if $ζ(Σ)\ge 4$ then Map($Σ$) has infinite coarse rank, and if $ζ(Σ)\ge 5$ then Map($Σ$) has at most quadratic divergence, hence is one-ended. |
| title | The non-peripheral curve graph and divergence in big mapping class groups |
| topic | Geometric Topology 20F65, 57K20 |
| url | https://arxiv.org/abs/2603.21560 |