Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.04813 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of 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 $\mathfrak{R}(λ,d)$.