The deformation space of non-orientable hyperbolic 3-manifolds

TL;DR

This paper studies the deformation space of non-orientable hyperbolic 3-manifolds, analyzing how their geometric structures can be deformed and how these relate to representation varieties, with applications to specific examples like the Gieseking manifold.

Contribution

It computes the deformation space of non-orientable hyperbolic 3-manifolds with ideal triangulations and explores the relationship with representation varieties near the holonomy.

Findings

Deformation spaces can differ from representation varieties when ends are non-orientable.

Explicit analysis of the Gieseking manifold illustrates the theoretical results.

Deformations may not always be realizable as pair deformations when non-orientability is involved.

Abstract

We consider non-orientable hyperbolic 3-manifolds of finite volume $M^3$. When $M^3$ has an ideal triangulation $\Delta$, we compute the deformation space of the pair $(M^3, \Delta)$ (its Neumann Zagier parameter space). We also determine the variety of representations of $\pi_1(M^3)$ in $\mathrm{Isom}(\mathbb{H}^3)$ in a neighborhood of the holonomy. As a consequence, when some ends are non-orientable, there are deformations from the variety of representations that cannot be realized as deformations of the pair $(M^3, \Delta)$. We also discuss the metric completion of these structures and we illustrate the results on the Gieseking manifold.