Topology of the space of $d$-pleated surfaces

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 $\mathfrak{R}(\lambda ,d)$.

69 pages, 17 figures. Comments are welcome!