Topology of the space of $d$-pleated surfaces

TL;DR

This paper investigates the topology of the space of $d$-pleated surfaces with a fixed geodesic lamination on a surface, revealing its structure as a real-analytic manifold with specific topological features.

Contribution

It characterizes the topology of the space of $d$-pleated surfaces, showing it is diffeomorphic to a product of Euclidean spaces, tori, and a cyclic group, and relates it to the broader representation space.

Findings

The space $rak{R}(l,l)$ is real-analytically diffeomorphic to a product of Euclidean space, torus, and cyclic group.

Each connected component of the conjugacy class space contains exactly one component of $rak{R}(l,l)$.

The topology of the space is explicitly described in terms of the genus and the dimension $d$.

Abstract

Given a maximal geodesic lamination $\lambda$ on a closed oriented surface $S$ of genus $g$, the space of $d$-pleated surfaces with pleating locus $\lambda$ is an open subset of $\mathrm{Hom}(\pi_1(S),\mathsf{PGL}_d(\mathbb{C}))$ obtained by applying generalized bending along $\lambda$ 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 $\mathfrak{R}(\lambda,d)$ of conjugacy classes of $d$-pleated surfaces with pleating locus $\lambda$. Firstly, we prove that $\mathfrak{R}(\lambda,d)$ is real-analytically diffeomorphic to $\mathbb{R}^{(d^2-1)(2g-2)}\times(\mathbb{R}/2\pi\mathbb{Z})^{(d^2-1)(2g-2)}\times \mathbb{Z}_d$, 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 $\mathrm{Hom}(\pi_1(S),\mathsf{PGL}_d(\mathbb{C}))$ contains exactly one component of $\mathfrak{R}(\lambda,d)$.