Generalized Teichm\"{u}ller space of non-compact 3-manifolds and Mostow rigidity

TL;DR

This paper studies a space of hyperbolic structures on certain 3-manifolds built from ideal tetrahedra, showing it is Euclidean and exploring whether these structures are uniquely determined by edge angles.

Contribution

It introduces a generalized Teichmüller space for non-compact 3-manifolds constructed from ideal tetrahedra and characterizes its topological and dimensional properties.

Findings

The space of hyperbolic metrics is homeomorphic to a Euclidean space.

The dimension of this space is explicitly computed.

Examples suggest edge angles may not always determine the structure uniquely.

Abstract

Consider a 3$-$dimensional manifold $N$ obtained by gluing a finite number of ideal hyperbolic tetrahedra via isometries along their faces. By varying the isometry type of each tetrahedron but keeping fixed the gluing pattern we define a space $\mathcal{T}$ of complete hyperbolic metrics on $N$ with cone singularities along the edges of the tetrahedra. We prove that $\mathcal{T}$ is homeomorphic to a Euclidean space and we compute its dimension. By means of examples, we examine if the elements of $% \mathcal{T}$ are uniquely determined by the angles around the edges of $N.$