The deformation space of nonorientable hyperbolic 3–manifolds

Durán Batalla, Juan Luis; Porti, Joan

doi:10.2140/agt.2024.24.109

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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

DOI: 10.2140/agt.2024.24.109

Abstract

We consider nonorientable hyperbolic $3$ –manifolds of finite volume $M^{3}$ . When $M^{3}$ has an ideal triangulation $Δ$ , we compute the deformation space of the pair $(M^{3}, Δ)$ (its Neumann–Zagier parameter space). We also determine the variety of representations of $π_{1} (M^{3})$ in $Isom (ℍ^{3})$ 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 $(M^{3}, Δ)$ . We also discuss the metric completion of these structures and we illustrate the results on the Gieseking manifold.

Keywords

three-manifold, hyperbolic Dehn filling, nonorientable

Mathematical Subject Classification

Primary: 57K32

Secondary: 57K35, 57Q99

References

Publication

Received: 22 February 2021

Revised: 4 April 2022

Accepted: 9 August 2022

Published: 18 March 2024

Authors

Juan Luis Durán Batalla
Departament de Matemàtiques Universitat Autònoma de Barcelona Barcelona Spain

Joan Porti
Departament de Matemàtiques Universitat Autònoma de Barcelona, and Centre de Recerca Matemàtica Barcelona Spain
https://mat.uab.cat/~porti/

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Volume 24, issue 1 (2024)