On dynamics of the Mapping class group action on relative $\text{PSL}(2,\mathbb{R})$-Character Varieties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929714200838144 |
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| author | Nair, Ajay Kumar |
| author_facet | Nair, Ajay Kumar |
| contents | In this paper, we study the mapping class group action on the relative $\text{PSL}(2,\mathbb{R})$-character varieties of punctured surfaces. It is well known that Minsky's primitive-stable representations form a domain of discontinuity for the $\text{Out}(F_n)$-action on the $\text{PSL}(2,\mathbb{C})$-character variety. We define simple-stability of representations of fundamental group of a surface into $\text{PSL}(2,\mathbb{R})$ which is an analogue of the definition of primitive stability and prove that these representations form a domain of discontinuity for the $\text{MCG}$-action. Our first main result shows that holonomies of hyperbolic cone surfaces are simple-stable. We also prove that holonomies of hyperbolic cone surfaces with exactly one cone-point of cone-angle less than $π$ are primitive-stable, thus giving examples of an infinite family of indiscrete primitive-stable representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09513 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On dynamics of the Mapping class group action on relative $\text{PSL}(2,\mathbb{R})$-Character Varieties Nair, Ajay Kumar Geometric Topology Metric Geometry In this paper, we study the mapping class group action on the relative $\text{PSL}(2,\mathbb{R})$-character varieties of punctured surfaces. It is well known that Minsky's primitive-stable representations form a domain of discontinuity for the $\text{Out}(F_n)$-action on the $\text{PSL}(2,\mathbb{C})$-character variety. We define simple-stability of representations of fundamental group of a surface into $\text{PSL}(2,\mathbb{R})$ which is an analogue of the definition of primitive stability and prove that these representations form a domain of discontinuity for the $\text{MCG}$-action. Our first main result shows that holonomies of hyperbolic cone surfaces are simple-stable. We also prove that holonomies of hyperbolic cone surfaces with exactly one cone-point of cone-angle less than $π$ are primitive-stable, thus giving examples of an infinite family of indiscrete primitive-stable representations. |
| title | On dynamics of the Mapping class group action on relative $\text{PSL}(2,\mathbb{R})$-Character Varieties |
| topic | Geometric Topology Metric Geometry |
| url | https://arxiv.org/abs/2502.09513 |