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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.04813 |
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| _version_ | 1866916884834680832 |
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| author | Maloni, Sara Martone, Giuseppe Mazzoli, Filippo Zhang, Tengren |
| author_facet | Maloni, Sara Martone, Giuseppe Mazzoli, Filippo Zhang, Tengren |
| contents | Given a maximal geodesic lamination $λ$ on a closed oriented surface $S$ of genus $g$, the space of $d$-pleated surfaces with pleating locus $λ$ is an open subset of $\mathrm{Hom}(π_1(S),\mathsf{PGL}_d(\mathbb{C}))$ obtained by applying generalized bending along $λ$ to Hitchin representations. When $d=2$, one recovers abstract pleated surfaces in $\mathbb{H}^3$. In this paper, we study the topology of the space $\mathfrak{R}(λ,d)$ of conjugacy classes of $d$-pleated surfaces with pleating locus $λ$. Firstly, we prove that $\mathfrak{R}(λ,d)$ is real-analytically diffeomorphic to $\mathbb{R}^{(d^2-1)(2g-2)}\times(\mathbb{R}/2π\mathbb{Z})^{(d^2-1)(2g-2)}\times \mathbb{Z}_d$, where $\mathbb{Z}_d$ denotes the finite cyclic group of order $d$. Furthermore, we show that each connected component of the space of conjugacy classes in $\mathrm{Hom}(π_1(S),\mathsf{PGL}_d(\mathbb{C}))$ contains exactly one component of $\mathfrak{R}(λ,d)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04813 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Topology of the space of $d$-pleated surfaces Maloni, Sara Martone, Giuseppe Mazzoli, Filippo Zhang, Tengren Geometric Topology Given a maximal geodesic lamination $λ$ on a closed oriented surface $S$ of genus $g$, the space of $d$-pleated surfaces with pleating locus $λ$ is an open subset of $\mathrm{Hom}(π_1(S),\mathsf{PGL}_d(\mathbb{C}))$ obtained by applying generalized bending along $λ$ to Hitchin representations. When $d=2$, one recovers abstract pleated surfaces in $\mathbb{H}^3$. In this paper, we study the topology of the space $\mathfrak{R}(λ,d)$ of conjugacy classes of $d$-pleated surfaces with pleating locus $λ$. Firstly, we prove that $\mathfrak{R}(λ,d)$ is real-analytically diffeomorphic to $\mathbb{R}^{(d^2-1)(2g-2)}\times(\mathbb{R}/2π\mathbb{Z})^{(d^2-1)(2g-2)}\times \mathbb{Z}_d$, where $\mathbb{Z}_d$ denotes the finite cyclic group of order $d$. Furthermore, we show that each connected component of the space of conjugacy classes in $\mathrm{Hom}(π_1(S),\mathsf{PGL}_d(\mathbb{C}))$ contains exactly one component of $\mathfrak{R}(λ,d)$. |
| title | Topology of the space of $d$-pleated surfaces |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2508.04813 |