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The deformation space of nonorientable hyperbolic $3$–manifolds

Juan Luis Durán Batalla and Joan Porti

Algebraic & Geometric Topology 24 (2024) 109–140

We consider nonorientable hyperbolic 3–manifolds of finite volume M3. When M3 has an ideal triangulation Δ, we compute the deformation space of the pair (M3,Δ) (its Neumann–Zagier parameter space). We also determine the variety of representations of π1(M3) in Isom(3) in a neighborhood of the holonomy. As a consequence, when some ends are nonorientable, there are deformations from the variety of representations that cannot be realized as deformations of the pair (M3,Δ). We also discuss the metric completion of these structures and we illustrate the results on the Gieseking manifold.

three-manifold, hyperbolic Dehn filling, nonorientable
Mathematical Subject Classification
Primary: 57K32
Secondary: 57K35, 57Q99
Received: 22 February 2021
Revised: 4 April 2022
Accepted: 9 August 2022
Published: 18 March 2024
Juan Luis Durán Batalla
Departament de Matemàtiques
Universitat Autònoma de Barcelona
Joan Porti
Departament de Matemàtiques
Universitat Autònoma de Barcelona, and Centre de Recerca Matemàtica

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