Dominating surface-group representations via Fock-Goncharov coordinates
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915079616724992 |
|---|---|
| author | Barman, Pabitra Gupta, Subhojoy |
| author_facet | Barman, Pabitra Gupta, Subhojoy |
| contents | Let $S$ be a punctured surface of negative Euler characteristic. We show that given a generic representation $ρ:π_1(S) \rightarrow \mathrm{PSL}_n(\mathbb{C})$, there exists a positive representation $ρ_0:π_1(S) \rightarrow \mathrm{PSL}_n(\mathbb{R})$ that dominates $ρ$ in the Hilbert length spectrum as well as in the translation length spectrum, for the translation length in the symmetric space $\mathbb{X}_n= \mathrm{PSL}_n(\mathbb{C})/\mathrm{PSU}(n)$. Moreover, the $ρ_0$-lengths of peripheral curves remain unchanged. The dominating representation $ρ_0$ is explicitly described via Fock-Goncharov coordinates. Our methods are linear-algebraic, and involve weight matrices of weighted planar networks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15378 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dominating surface-group representations via Fock-Goncharov coordinates Barman, Pabitra Gupta, Subhojoy Geometric Topology Let $S$ be a punctured surface of negative Euler characteristic. We show that given a generic representation $ρ:π_1(S) \rightarrow \mathrm{PSL}_n(\mathbb{C})$, there exists a positive representation $ρ_0:π_1(S) \rightarrow \mathrm{PSL}_n(\mathbb{R})$ that dominates $ρ$ in the Hilbert length spectrum as well as in the translation length spectrum, for the translation length in the symmetric space $\mathbb{X}_n= \mathrm{PSL}_n(\mathbb{C})/\mathrm{PSU}(n)$. Moreover, the $ρ_0$-lengths of peripheral curves remain unchanged. The dominating representation $ρ_0$ is explicitly described via Fock-Goncharov coordinates. Our methods are linear-algebraic, and involve weight matrices of weighted planar networks. |
| title | Dominating surface-group representations via Fock-Goncharov coordinates |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2405.15378 |